INFINITESIMAL CALCULUS Historical Introduction.
THE mathematical and physical sciences owe their present great development to the introduction of the infinitesimal calculus. The power, for example, of that calculus as an instrument of analysis has vastly extended the science of geometry, so that the investigations of the ancient Greeks go but a short way into the field of knowledge which has been laid open by the modern method; the discoveries of Archimedes and Apollonius are now easy deductions from its more extended results.
So long as the early geometers confined their speculations to the comparison of the areas of rectilinear figures they encountered little difficulty. They readily showed that the determination of the area of any such figure can be reduced to that of a rectangle, or of a square, and thus be completely effected. This process of finding areas was named the “method of quadratures.” It failed, however, when they attempted to determine the areas bounded by curved lines, or the surfaces of the elementary solids such as the right cone and the sphere. In treating of these the ancients found it necessary to introduce new notions and modes of demonstration into geometry, and the difficulty of comparing the areas of curvilinear with those of rectilinear figures gave rise to the “method of exhaustions.” The fundamental principle of this method consists in conceiving the continual approach of two varying magnitudes to a fixed intermediate magnitude, with which they never become identical, though they may approach it to within less than any assignable difference. For example, a polygon may be inscribed in a circle, and another circumscribed to it, each differing from it by less than any assignable area; hence the ancients may have concluded that areas of circles have to each other the same ratio as the similar polygons inscribed in or circumscribed to them, —that is, the ratio of the squares of the radii. But, as this kind of proof was of a different nature from that by which the more elementary doctrines were established, the Greek geometers fortified it by a reductio ad absurdum, — proving, in the above example, that the square of the radius of one circle is to that of another as the area of the former is to a space which is neither less nor greater than the latter, and therefore exactly equal to it.
By the aid of this method Archimedes arrived at his great geometrical discoveries. He determined that the ratio of the circumference to the diameter of a circle lies between 3∣ and 3j-⅞, by considering the regular polygons of ninety-six sides which may be inscribed in or circumscribed to the circle. He proved that the area of a segment of a parabola cut off by any chord equals two-thirds of a parallelogram included between the chord and the parallel tangent to the curve. He determined the quadrature of the ellipse. In the curves named after him the “spirals of Archimedes,” he showed how to draw a tangent at any point, and also determined the area of any portion.
In space of three dimensions, Archimedes proved that the surface of a sphere equals four times that of one of its great circles, that the surface of a spherical cap is equal to the area of a circle the length of whose radius is the distance from the vertex of the cap to any point on its bounding circle; that a sphere has a volume which is two-thirds of that of a cylinder circumscribed to it, and that their surfaces are in the same ratio. Further, the same method of exhaustions furnished Archimedes with the cubature of conoids and spheroids, as he termed surfaces generated by the revolution of the parabola, the hyperbola, and the ellipse.
During nearly two thousand years no new method enabled mathematicians to rise to a higher generality than that attained in the works of the great Greek geometers. The celebrated Kepler was the first to extend the results of Archimedes. In his treatise entitled Nova Stereornetria Doliorum ; accessit stereometriae Archimedeae supplementum (1615), he discussed a number of solids of revolution,— for example, those formed by the revolution of a conic section about any ordinate, or a tangent at the vertex, or any line within or without the curve. Thus he considered some ninety new solids, and proposed problems concerning them; of these problems he resolved only a few of the most simple. In this treatise he introduced for the first time the name and notion of “infinity” into the language of geometry. Thus, he considered a circle as composed of an “infinite” number of triangles, having their common vertex at the centre, and forming the circumference by their bases. In like manner he regarded a cone as composed of an infinite number of pyramids, having their vertices at its vertex, and standing on an infinite number of triangular bases, bounded by the circular base of the cone. It may also be noted that Kepler was the first to observe that the increment of a variable—the ordinate of a curve, for example—is evanescent for values infinitely near a maximum or minimum value of the variable. This remark contains the germ of the rule for determining “maxima” and “minima,” given by Fermat about twenty years subsequently.
Several years after Kepler had given his method of determining volumes of revolution, another celebrated theory, of a similar kind,—the “geometry of indivisibles” (1635) of Cavalieri, professor of geometry at Bologna,— marks an epoch in the progress which science has made in modern times. In this work lines were considered as composed of an infinite number of points, surfaces of lines, and solids of surfaces. For example, if the perpendicular of a triangle be divided into an infinite number of equal parts, and through each point of division a line be drawn parallel to the base and terminated by the sides, then, according to Cavalieri, we may consider the area of the triangle as the sum of all these parallel lines, regarded as its elements. Again, as these parallels form a series in arithmetical progression, of which the first term is zero, this sum is represented by half the product of the last term and the number of terms. Now the base js the last term, and the altitude measures the number of terms; thus he deduced the ordinary expression for the area of a triangle. Cavalieri applied his method to a number of problems, such as finding the volumes of pyramids, the areas of certain simple curves of the parabolic species, the determination of centres of gravity, &c.; and it is remarkable that he was the first who gave an accurate demonstration of the well-known properties of the centre of gravity, originally announced by Pappus, but commonly called Guldin’s theorems. It is accordingly to Cavalieri, and not to Guldin, that the credit is due of having made the first advance beyond Pappus. Cavalieri’s method is analogous to that employed in the integral calculus, tl∣e “indivisible being that which has since been styled the “differential element” of the integral.
The method of Cavalieri was severely criticized "by some of his contemporaries, more especially by Guldin. They alleged that, since a line has no breadth, no number of right lines, however great, when taken together, could make up a plane area. This objection was answered by Cavalieri; but the reply was stated in the clearest form by Pascal, who observed (letter to M. de Carcavi, 1658) that the method of indivisibles possessed all the rigour of that of exhaustions, from which it differed only in the manner of expression, and that, when we conceive an area as a sum of a system of parallel ordinates, we mean in reality an indefinite number of rectangles under the several ordinates, and the small equal portions into which we conceive the common perpendicular to these ordinates to be divided. This passage is remarkable—as was well observed by Carnot—as it shows that the notion of mathematical infinity, as now employed, was not strange to the geometers of that time ; for it is clear that Pascal employed the word “indefinite” in the same signification as we now attach to the word “infinite,” and that he called “small” that which is now called “infinitely small,” also that he neglected these small quantities in comparison with finite quantities—thus he regarded as simple rectangles the small portions of the area of the curve comprised between two consecutive ordinates, neglecting the small triangles which have for their bases the differences of these ordinates. Carnot adds that no person attempted to reproach Pascal with want of rigour in his demonstrations.
Pascal applied the method of Cavalieri with eminent success to the investigation of properties of the cycloid, and other problems. His researches, according to D’Alembert, closely approach to the integral calculus, and form the connexion between the methods of Archimedes and of Newton.
The most important application, however, of Cavalieri’s method was that of Wallis, Savilian professor at Oxford, who, in 1655, gave an admirable specimen of this method in his Arithmetica Infinitorum, sive nova Methodus inquirendi in Curvilinearum Quadraturam. Pursuing Cavalieri’s views, he reduced the problem of finding the areas of a large number of curves, and the volumes of solids of revolution, to the summation of the powers of the terms of arithmetical series, consisting of an infinite number of terms,—or rather to the determination of the ratio of the arithmetical mean of all such powers of the terms to the like power of the last term.
For example, in the series of square numbers 0, 1, 4, 9, 16, &c., the ratio of the mean to the last is, for the first , κ , 0+1+4 i 1 1 f . 1 n . f 0+l+4 + 9
three terms, = ⅜ + ⅛ for the first four, 9+9+9+9 ι,ι r 4-i /·4.c 0 + 1 + 4+9 + 16 , ,
= ⅞ + ⅛; for the tot five, nj + i6- l -18 + 16 + i6 = * + ⅛ in like manner the next fraction is ⅜ + ⅛. Hence Wallis noticed that the fractions approach nearer and nearer to ⅜; and, as the denominators in the fractions ∙⅛, -j⅛-, -f- ι , π ⅛ form an arithmetical series, with a common difference 6^ it follows that, when the number of terms is indefinitely increased, the resulting fraction becomes ultimately ⅜.
Wallis applied the same method to the series 0, 1 3 , 2 3 , 3 3 , &c., and found without difficulty that the aforesaid ratio is I in this case; and so generally. He also introduced into analysis the notation of fractional indices instead of radicals, and extended his method of summation to series proceeding by fractional powers of the natural numbers 1, 2, 3, &c. Wallis was enabled by these principles to obtain the areas of many curves, and the volumes of solids which had not been previously found. He also, by aid of this method, combined with the principles of “interpolation,” arrived at his well-known expression for τr, viz. :—
π 2+ £6 fi;8
4 - 3∙3 ' 5∙5 ‘ 7∙7
Again, in his treatise De Curν. rectif. (1659), Wallis showed that certain curves were capable of being “rectified,” or that straight lines might be found to which they were exactly equal, a remark which was very soon verified by a young English mathematician William Neil, who, by Wallis’s method, obtained in 1660 the length of any arc of a semicubical parabola. This is the first curve that was rectified. The cycloid is the second j its rectification was effected by Sir C. Wren (Phil. Trans., 1673). The methods we have thus far considered were more especially precursory to the integral calculus, having mainly reference to the quadrature of curves and cubature of solids. We now propose to consider the question of tangents to curves, in which the differential calculus may be said to have originated.
The great discovery of Descartes in his application of algebra to geometry (1637) imparted to the latter science the character of abstraction and generality which distinguishes modern from ancient geometry. By it the study of curves was brought under the domain of analysis, and instead of investigation being restricted to particular properties of a few isolated curves, as it had been hitherto, general views and methods applicable to all curves were introduced.
Hence the general problem of drawing tangents to curved lines started immediately into prominence. It was found necessary to depart from the definition of tangents given by the ancient geometers, and to consider them in other points of view. A tangent, accordingly, came to be regarded either (1) as a secant of which the points of intersection became coincident; or (2) as the prolongation of the element of the curve, regarded as a polygon of an infinite number of sides; or (3) as the direction of the resultant motion by which the curve may be described. The first view was that of Descartes and Fermat; the second was introduced by Barrow, who thus simplified the method of Fermat; and the third was that of Roberval.
Descartes’s method of drawing a tangent consisted in supposing a circle (whose centre he placed on the axis of x) to cut the curve in two points; then, if the radius of the circle be supposed to decrease, its centre remaining fixed, so that the points of section approach nearer and nearer and finally coincide, the circle will touch the curve; thus, by aid of the equation of the curve, the problem was reduced to one of finding the condition of equal roots in an equation. This method is remarkable as being the first general process of applying analysis to the problem of tangents; at the same time it is only capable of practical application in a small number of simple cases.
Many years subsequently (Act. Erud. Lips., 1691) John Bernoulli extended Descartes’s method with success to the problem of finding the centre of curvature and the equation of the evolute of an algebraic curve. In his application he supposed the centre of a circle taken on the normal to a curve, and the centre to vary until three of the points of intersection of the circle with the curve became coincident, i.e., so that the resulting equation should have three equal roots. Thus, for example, he showed, without difficulty, that the evolute of a parabola was a semicubical parabola. He also remarked that, when four roots coincide, the centre of curvature becomes a cusp on the evolute.
It should also be noticed that we owe to Descartes the general method of drawing a tangent to a roulette. This was given by him in a letter to Mersenne (Aug. 23, 1638), from which we take the following extracts :—“I have been very glad to see the questions which you say that the geometers, even Μ. Roberval, whom you esteem the principal of them, confess that they cannot solve; for in investigating them I may discover whether my analysis is better than theirs. The first of these questions is that of drawing tangents to curves described by a roulette motion. My solution is as follows. If a rectilinear polygon be conceived to roll on a right line, the curve described by any one of its points will be composed of a number of arcs of circles, and the tangent at any point on one of these arcs is perpendicular to the line drawn from the point to that in which the polygon is in contact with the base, when describing the arc. Consequently, if we consider a rolling curve as a polygon of an infinite number of sides, we see clearly that the roulette traced by any point must possess the same property; that is to say, the tangent at any of its points is perpendicular to the right line connecting it with the point of contact of the rolling curve and its base.” In this we perceive that Descartes gave a genuine and most important application of the infinitesimal method.
Again, Descartes first introduced the method of indeterminate coefficients into analysis,—a principle, as was ably shown by Carnot, which is of itself sufficient to establish, by ordinary algebra, the fundamental principles of the infinitesimal calculus.
The method of Fermat for drawing tangents was based on his method of maxima and minima. This latter was founded, as already observed, on a principle of Kepler’s, viz., that, whenever a magnitude attains a maximum or minimum, its increment or diminution, for a very small change in the variable on which it depends, becomes evanescent.
Accordingly, to determine the maximum or minimum of any function of x, Fermat substituted x + e instead of x, and equated the two consecutive values of the function; then, removing the common terms, and dividing by e, , he made e = 0, and obtained an equation for determining the maximum or minimum value.
Thus, adopting the modern notation, let y = fix), and y 1 = f(x + e), then f(x + é) -f(x) = 0. Dividing by e, f(X + e)-fX) (y. c ’ hence f'(x) = 0.
Thus the roots of the derived equation, f'(x) = 0, correspond to the maximum or minimum values of f(x). Consequently we see that Fermat’s rule agrees with that of the differential calculus, and in fact is the method of the calculus as applied to such cases.
In consequence of Fermat’s both having introduced the conception of an infinitely small difference, and also having arrived at the principle of the calculus for determining maxima and minima, it was maintained by Laplace, Lagrange, Fourier, and other eminent French mathematicians that Fermat ought to be regarded as the first inventor of the differential calculus. In reply to this we need but introduce the remark of their distinguished countryman Poisson, “that this calculus consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems” ( Mém.de l’Àcad. des Sci., 1831).
Fermat seems to have given no general demonstration of his method, but contented himself with giving particular applications of it to some problems of maxima and minima, as well as to finding the tangents to and the centres of gravity of a few curves.
Fermat applied his method to drawing a tangent, as follows :—
Suppose CD (fig. 1) the ordinate, and CF the tangent at the point C in a curve, meeting the axis AB in F; from E, a near point on CF, draw an ordinate EG; then CD FC HC
τ - τi = FΓr, ≥ tvs > *f the curve be concave to the axis, l√f Gr trΓ
CD HG and —- < —£ , if the curve be convex.
JDK Gr
Hence, in either case, the ratio of the ordinate CD to the subtangent DF is a maximum or a minimum relatively to the ratio for a near ordinate HG to GF, the abscissa measured from F, the foot of the tangent.
Accordingly, if CD = y, HG = y 1 , and DF = i, we have, by the method of maxima and minima, t tic
It is easily seen that this method furnishes the ordinary value for the subtangent, as obtained by infinitesimals; for, denoting by x, y the coordinates of C, let Z = FD, DG = <⅛ y 1 =y + dy, and we have y y+dy . t≈yt t + dx ’ ’ j dy from which the subtangent t can be obtained.
The method of Fermat was improved and extended by an Italian, Cardinal Ricci, in his Geometrica exercitatio (1666). Ricci was the first who showed that, if (<ι - ar) m ir n is a maximum, we must have nα x= .
m+n
This he easily established when m and n are integers, from the principle that if a magnitude be divided into r equal parts, their continued product is greater than that obtained by dividing it into r parts in any other manner
The following application, as given by him, to the curve y m =2⅛r n , m>n, will help to illustrate this method of drawing tangents. To draw the tangent at C (fig. 1 ) take AF: AD = m - n: n, and join FC; then FC touches the curve at C.
For the product AF m ^ n AD n is a maximum by the preceding lemma; hence the product AF" -,, AG" is not a maximum for the line FG; consequently
AF ,ft - n AD" AF m -"AG n . ∕ADV∙ ∕FD∖"∙ .
FD” ^ FG” ’ ’ ∖AG√ ∖FG∕ ’
but, from the equation of the curve,
/ADV ∕CD∖ m
(ag∕ (hg∕ ’
a . FD CD CD CD λ ,, p ∩ pττ also FS-Bδ∙ H3>eS' oteγ1 =∙ gπ ' i.β., the point E falls outside the curve. In like manner it can be shown that any other point on CF lies outside the curve, and consequently CF touches the curve at C.
Barrow, Newton’s predecessor in the Lucasian chair of mathematics at - Cambridge, simplified and extended the method of Fermat, and aclvanced a step further in the development of the infinitesimal method, by the introduction of two infinitesimals instead of one in the problem of drawing a tangent. His method was as follows :—Let x, y be the coordinates of a point P on a curve (fig. 2), and take Q an adjacent point; let e = PR = MN be the increment of x, and α = QR the increment of y ; then, substituting x + e for x, and y + a for y, in the equation of the curve, subtracting the equation of the curve for the original values, and rejecting all terms of the second and higher degrees in a and e, he obtained the limiting value of a: e, or of PM: MT, thus determining the value of the subtangent. The triangle PQR, which has for its sides the elements of the curve, of the abscissa, and of the ordinate j has been called Barrow’s differential triangle.
The elements which Barrow represented by a and e Leibnitz subsequently styled dy and dx, the differentials of the ordinate and 'abscissa of the point on the curve. Thus Barrow anticipated the methods of Leibnitz and Newton of drawing tangents, so far as rational algebraic curves were concerned. Barrow’s researches were delivered in his professorial lectures in 1664,1665, and 1666, and were published in 1670, under the title of Lectiones Mathematicae.
The method of tangents of Roberval is based on the conception of the composition of motions, recently introduced by Galileo into mechanics, and depends on finding, from the properties of the curve, the different components of the motion of the point at which the tangent is required. The direction of the resultant of these motions determines that of the tangent. This method bears an analogy to Newton’s method of “fluxions,” but is very limited in its application on account of the impossibility of applying it except in a few cases. Roberval applied it successfully to the following curves—the parabola, hyperbola, ellipse, conchoid of Nicomedes, limaçon of Pascal, spiral of Archimedes, quadratrix, cissoid, cycloid, companion to the cycloid, and the parabola of Descartes.
We thus see that both in England and on the Continent the principles of the infinitesimal calculus were being gradually developed. Their importance was seen and understood, and they were employed in extending the dominion of geometry. Nothing more was required but an appropriate notation to form them into a system. This was accomplished by Newton and Leibnitz, who appeared nearly at the same time in the field of discovery. In ascribing to each of these great men the full honour due to the merit of the invention of the calculus, it is proper to add that this is a question which at one time divided the opinions of the scientific world, and gave rise to a controversy which was agitated with great keenness for almost a whole century. There never could be any doubt as to Newton being the inventor of the calculus of fluxions; but the question strongly contested has been, whether Leibnitz invented his calculus independently, or borrowed it from the lluxional calculus, with which at bottom it is identical.
Leibnitz, born in 1646, four years after Newton, was also later than Newton in beginning his career of discovery in mathematics. In 1673, being in London, he communicated to some members of the Royal Society what he supposed to be discoveries relative to the differences of numbers. It was, however, shown to him that the same subject had been previously discussed by Mouton, a French geometer. He then appears for the first time to have turned his attention to infinite series. On his return to Germany in 1674 he announced to Mr Oldenburg, secretary to the Royal Society, that he possessed very general analytical methods, depending on infinite series, by which he had found theorems of great importance relating to the quadrature of the circle. In reply Oldenburg informed him that Newton and Gregory had discovered similar methods for the quadrature of curves, which extended to the circle.
On June 13, 1676, Newton addressed a letter to Oldenburg, for transmission to Leibnitz. It contained his binomial theorem, the now well-known expressions for thq expansion of an arc in terms of its sine, and the converse, that of the sine in terms of the arc. Contrary to modern procedure, he deduced the latter from the former, by the method of reversion of series,—a method called by Newton the “extraction of roots.” This letter also contained an expression in an infinite series for the arc of an ellipse, and various other results, accompanied, however, only by brief indications of his method of demonstration.
On the 27th of the following August, Leibnitz sent a reply through Oldenburg, in which he requested fuller explanation from Newton of his theorems and method of demonstration. Leibnitz added that he possessed another method of extensive application in geometry and mechanics, of which he gave some examples.
To this communication Newton replied on October 24, 1676, in a letter which occupies thirty pages in Newton’s Opuscula (ed. Cast.). As this letter probably gives a more complete account of the order and dates of Newton’s discoveries than is to be obtained elsewhere, it appears desirable to give a brief abstract of it here. He commences by commending the very elegant method of Leibnitz for the treatment of series. He goes on to state that he himself had three methods of such treatment. His first was arrived at from the study of the method of interpolation of series by which Wallis had arrived at expressions for the area of the circle and hyperbola. Thus, by considering the series of expressions (1 - x 2 )⅛, (1 - x 2 )⅛, (1 - x 2 )⅛, (1 -ic 2 )⅜, (1 - x 2 )*, &c., he deduced from the known values of the alternate expressions, by the method of interpolations, the law which connects the successive coefficients in the expansions of the intermediate terms (1 - ic 2 )⅛, (1 - x 2 )∖ (l-x 2 )*, &c. Newton thus determined the area of the circle and hyperbola, in infinite series. He adds that this method would have completely escaped his memory if he had not a few weeks previously found the notes he had formerly made on the subject.
By following out the idea thus suggested, he was led to the discovery of his binomial expansion. This he tested in the case of (1-τ 2 )⅛ by the algebraic process of extracting the square root,—as also, in other cases, by direct multiplication. Having established this result, he was enabled to discard the method of interpolation, and to employ his binomial theorem as the most direct method of obtaining the areas and arcs of curves. Newton styled this his second method. He states that he had discovered it before the plague (in 1665-66) had compelled him to leave Cambridge, when he turned his attention to other subjects. He goes on to say that he had ceased to pursue these ideas as he suspected that Nicholas Mercator had employed some of them in his Logarithmotechwia (1668); and this led him to think that the remainder would have been found out before he himself was of sufficiently ripe age to publish his discoveries (priusquam ego aetatis essem maturae ad scribeιι∙ dum).
Newton proceeds to state that about 1669 he communicated through Barrow to Collins a compendium of his method subsequently called the “method of fluxions,” with applications to areas, rectification, cubature, &c. In his letter, however, he gave no explanation of this method, carefully concealing its nature in an anagram of transposed letters, thus—6α ce d ae 13e ff 7 i 3l 9n 4o 4g rr 4s 9i 12v x.
At the end of his letter Newton alludes to the solution of the “inverse problem of tangents,” a subject referred to iu Leibnitz’s letter. For the solution of such problems he says he has two methods, which also he disguises under an anagram. The meaning of this anagram is given in his Opuscula, and, as it throws light on Newton’s method of discovery, it is introduced here :—“Una methodus consistit in extractione fluentis quantitatis ex aequatione simul involvente fluxionem ejus. Altera tantum in assumptione seriei pro quantitate qualibet incognita ex qua caetera commode derivari possunt, et in collatione terminorum homologorum aequationis resultantis, ad eruendos terminos assumptae seriei.”
On June 21, 1677, Leibnitz sent a reply to Newton, through Oldenburg. In this he explained his method of drawing tangents to curves, introducing his notation, dx and dy, for the infinitely small differences of the successive coordinates of a point on the curve, and showed that his method could be readily applied if the equation contained irrational functions. Further on he gave one or two examples of the inverse method of tangents, such as to find the curve whose subtangent is b + cy + df 2 ∙ - x. This, which is a problem involving the integration of a differential equation of the first order, shows that Leibnitz was then in possession of the principles of the integral calculus. The sign of integration has been found to have been employed by him in a manuscript of 29th October 1675, preserved in the royal library of Hanover (Gerhardt, Die Entdeckung der höheren Analysis, 1855). This date is of importance, as it proves conclusively that Leibnitz was in possession of his method before he had received through Oldenburg any account of Newton’s method of fluxions, and thus shows how unfounded was the statement made in the Commercium Epistolicum that Leibnitz had borrowed his calculus from Newton.
The death of Oldenburg, which took place shortly afterwards, put an end to this correspondence. In the year 1684 Leibnitz, for the first time, made his method public, in the Acta Eruditorum of Leipsic, under the following title, “Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus.” Newton’s method did not appear until 1687, when he published it, in a geometrical form, as the method of prime and ultimate ratios, in his great work Philosophise Naturalis Principia Mathematica ; consequently, while Newton’s claim to the priority of discovery is now admitted by all, it is no less certain that Leibnitz was the first to publish his method. It is also certain that Leibnitz enjoyed unchallenged for fifteen years the honour of being the inventor of his calculus; even Newton himself rendered him that justice in the first two editions of his Principia.
Subsequently, however, a foreigner, Fatio de Duillier, piqued, as is abundantly manifested in his tract, at having been omitted in an enumeration by Leibnitz of eminent geometers alone capable of solving John Bernoulli’s celebrated problem of the line of quickest descent, published in 1699, at London, a memoir on the problem. In this he declared that he was obliged by the undeniable evidence of things to acknowledge Newton, not only as the first, but as by many years the first inventor of the calculus, from whom, whether Leibnitz, the second inventor, borrowed anything or not, he would rather they who had seen Newton’s letters and other manuscripts should judge than himself.
This insinuation drew forth an animated reply from Leibnitz, in the Acta Eruditorum, May 1700, in which he cited Newton’s letters, as also the testimony which Newton had rendered to him in the Principia, as proof of his claim to an independent authorship of his method. A reply was sent by Duillier, which the editors of the Acta Eruditorum refused to publish (quasi lites aversati). Here the dispute rested for a time. It was revived in the year 1705, when, on the publication of Newton’s Tractatus de Quadratura Curvarum, an unfavourable review of the work,—written by Leibnitz, as has since been established,—appeared in the Acta Eruditorum. In this review, among other observations, it was stated that Newton employed and had always employed fluxions instead of the differences of Leibnitz, just as Fabri had substituted, in his synopsis of geometry, motion instead of the indivisibles of Cavalieri. This statement excited great indignation among British mathematicians, one of whom—Keill, Savilian professor of astronomy at Oxford—in a letter printed in the Philosophical Transactions of 1708, affirmed that Newton was, without doubt, the first inventor of the calculus, and that Leibnitz, in the Acta Eruditorum, had merely changed the name and the notation. Leibnitz, thus directly charged with having taken his calculus from Newton, addressed a letter, March 1711, to Mr (afterwards Sir Hans) Sloane, the secretary of the Royal Society, in which he reminded him that, a similar accusation having been made some years previously by M. Fatio de Duillier, the Society and Newton himself had disapproved of it, and he requested the Society to require that Keill should retract his accusation. This Keill refused to do, and in answer addressed a letter of great length to Sloane, in which he professed to show, not only that Newton had preceded Leibnitz in the invention, but that he had given Leibnitz so many indications of his calculus that its nature might have been easily understood by any man of ordinary intelligence. That this was in substance the statement of Newton himself appears from the minutes of the Royal Society (of which he was president), April 5, 1711, in which it is stated “that the president gave a short account of the matter, referring to some letters, published by Dr Wallis, upon which Mr Keill was desired to draw up an account of the matter under dispute and set it in a just light.” Keill accordingly wrote a letter which was submitted to the Society on May 24.
This letter was forwarded to Leibnitz, who, on December 29, 1711, addressed a second letter to Sloane, requiring the Society to stop these unjust attacks of Keill, and saying that Keill was too young a man to know what had passed between Newton and himself. In conclusion, he submitted the matter to the equity of the Royal Society, and stated that he was persuaded that Newton himself would do him justice. The Society, thus appealed to, appointed a committee on 6th of March 1712, to examine the old letters and other documents which had passed between mathematicians on the subject and to furnish a report to the Society. The members of the committee, as originally appointed, were Arbuthnot, Hill, Halley, Jones, Machin, and Burnet. To these Robarts, a contributor to the Transactions, was added on the 20th; Bonet, the Prussian minister, on the 27th; and De Moivre, Aston, and Brook Taylor on the 17th of April. The complete list of the committee was not made public until.the question was investigated by the late Professor De Morgan, in 1852.
Their report, made on April 24, 1712, concluded as follows :—“The differential method is one and the same with the method of fluxions, excepting the name and mode of notation; Mr Leibnitz calling those quantities differences which Mr Newton calls moments or fluxions, and marking them with the letter d, a mark not used by Mr Newton. And therefore we take the proper question to be, not who invented this or that method, but who was the first inventor of the method; and we believe that those who have reputed Mr Leibnitz the first inventor, knew little or nothing of his correspondence with Mr Collins and Mr Oldenburg long before; nor of Mr Newton’s having that method above fifteen years before Mr Leibnitz began to publish it in the Acta Eruditorum of Leipsic. For which reasons, we reckon Mr Newton the first inventor, and are of opinion that Mr Keill, in asserting the same, has been no ways injurious to Mr Leibnitz.” On the same day the Society ordered the collection of letters and manuscripts, together with the report of the committee, to be printed, along with any other matter which would throw light on the question. This was accordingly done in the course of that year, under the title Commercium Epistolicum D. Johannis Collins et aliorum de analysi promota, jussu Societatis Pegiae in lucem editum, but not at first for general publication, the few copies printed being distributed as presents. In 1715 an elaborate account of the contents of this report was published by order of the Royal Society in their Transactions. The manuscript of almost the whole of this account has in recent years been found in Newton’s own handwriting. (Brewster’s Life of Newton, vol. ii. p. 75.)
In 1722 what is usually considered the second edition was published. The latest and most important edition is that of M. Biot and M. Lefort, published in Paris in 1856, in which many additional letters and documents necessary for an impartial appreciation of the question are added.
It would occupy too large a share of our space to detail the long and bitter controversy to which the Commercium Epistolicum gave rise. It suffices to state that from the time of its publication until long after the death of Leibnitz (November 14, 1716), and of Newton (March 28, 1727), this controversy was carried on, first between Newton and Leibnitz, and afterwards by their respective admirers. The feeling which induces men to exalt their own nation at the expense of their neighbours contributed immensely to increase the bitterness of the dispute. It is the less necessary nowadays to enter into the merits of this great quarrel, inasmuch as it has long been agreed on, by all mathematicians who have examined into the controversy, that Newton and Leibnitz are both justly entitled to be regarded as independent discoverers of the principles of the calculus, and that, while Newton was certainly master of the method of fluxions before Leibnitz discovered his method, yet Leibnitz had several years priority of publication.
The dispute seems, however, to have had a very injurious effect on the progress of mathematics in England; for, partly owing to the natural veneration for the lofty genius of Newton, but mainly, it would appear, in consequence of the strong national prejudice produced by the bitterness of the above-mentioned controversy, British mathematicians, for considerably more than a century, failed to perceive the great superiority of the notation originated by Leibnitz to that which Newton introduced. And thus, while the Bernoulli's, Euler, D’Alembert, Clairaut, Lagrange, Laplace, Legendre, and a number of other eminent Continental mathematicians were rapidly extending knowledge, by employing the infinitesimal calculus in all branches of mathematics, pure and applied, and producing a number of great treatises in every department, in England comparatively little progress was made.
In fact it was not until 1815—when three Cambridge graduates, who each afterwards rose to great distinction, Sir John Herschel, Babbage, and Peacock, published a translation of Lacroix’s smaller treatise on the calculus—that the algorithm universally adopted by Continental writers was introduced into the studies of the British universities. The great superiority of Leibnitz’s system of notation was soon acknowledged, and thus an immense impetus given to the study of mathematics, in all its branches. Ever since that time the method of fluxions, as a distinct method, has become almost obsolete; and it is now strange to read Newton’s own assertion in the preface to the Commercium Epistolicum, in which he claims that the method of fluxions is more elegant, more natural, more geometrical, more useful, more certain, and incomparably more universal, than that of Leibnitz.
We next proceed to give a brief account·of the notation and principles of the method of fluxions, as that which was first discovered.
The idea of a fluxion, as its name indicates, originated from that of motion, and all geometrical magnitudes were considered by Newton as capable of generation by continuous motion. Thus lines are conceived as generated by the motion of points, surfaces by that of lines, solids by surfaces, &c. Again, if we conceive a moving point as describing a curve, and the curve referred to coordinate axes, then the velocity of the moving point can be decomposed into two others, one parallel to the axis of x, the other to that of y ; these velocities are called the “fluxions” of x and y respectively, and the velocity of the point is the fluxion of the arc. Reciprocally, the arc is called the “fluent” of the velocity with which it is described; and the ordinates x and y are the fluents of their velocities respectively. Again, if the velocity of the moving point be regarded as constant, the fluxions of the abscissa and ordinate of any point on the curve (except in the case of a right line) will be variable; and their ratio at each instant will depend on the nature of the curve, t.e., on the relation between the coordinates. Reciprocally, the relation between the coordinates depends necessarily on that which exists at each instant between their fluxions. Hence we may seek to determine the relation between the fluxions, when we know that which exists between the coordinates, i.e., the equation of the curve; and reciprocally we may seek to discover the relation between the coordinates when we know that between their fluxions, either alone or combined with the coordinates themselves. The first part of the problem is called the “method of fluxions,” and the second the “inverse method of fluxions.”
Again, in the same case, not only do the coordinates x and y change, but also the subtangent, normal, radius of curvature, &c.; that is to say, each of these quantities increases or decreases more or less rapidly, as well as the coordinates themselves. All these quantities, accordingly, have fluxions, whose ratios are also determined by the motion of the point. Consequently these quantities may in like manner be regarded as “fluents.” Similar remarks apply to areas and surfaces regarded as fluents. Newton observes that he does not consider the time formally (formaliter), but supposes that one of the proposed quantities increases equably (aequahili fluxu), to which the others are referred (tanquam ad tempus). This fluent may be chosen at pleasure, and is what we now are accustomed to call the independent variable.
Again, if any quantities, regarded as fluents, be represented by letters, such as u, x, y, z, &c., the corresponding fluxions are represented by u, x, y, 2, &c., respectively. Next, if u, x, y, z be regarded as variable or fluent quantities, their fluxions are represented by ü, x, y, z, and are the fluxions of the fluxions of u, x, y, &c;, i.e., the second fluxions. If one of these, x for instance, be taken as the “principal fluxion,” then x is a constant, and consequently x = 0. In like manner we may have third fluxions, as well as those of higher orders.
Again, u, x, y, &c., may be regarded as themselves the fluxions of other quantities called their fluents. These quantities were represented by Newton, sometimes by u', x, y', &c., in other places by [w], [.r], &c. ; and from them it may be desired to proceed to the fluents.
Newton remarks that this second general problem involves three cases:—(1) when the. equation contains the fluxions of two quantities and but one of their fluents; (2) when the equation involves both the fluents as well as both the fluxions; (3) when the equation contains the fluents and the fluxions of three or more quantities.
The problem of finding the fluent when the fluxion is known is the simplest case of the first class, and is the same as the method of integration of Leibnitz. It was usually styled in Newton’s time the method of quadratures, for it is reducible to the problem of finding the area of a curve, since it can be easily seen that the fluxion of an area is the ordinate, when the abscissa is taken as the principal fluent. The second class comes under what is now called the solution of differential equations; this was styled in Newton’s time the “inverse method of tangents.” Newton’s third class is now treated of under the solution of “partial differential equations.”
The infinitely small parts by which the variable quantities increase in an indefinitely small time were called by Newton the “moments” of the fluent quantities; thus,he represented an infinitely small portion of time, called a moment, by o; then the moments or infinitely small increments of u, x, y, <fcc., are represented by uo, xo, ÿo, &c.; so that if u, x, y, &c., denote the values of the fluents at any instant, their values at the end of an indefinitely small interval of time are represented by u + uo, x + xo, &c.
For instance, let the fluents x, y, be connected by the equation x 3 -ax 2 + axy-y 3 =0; then, substituting x+xο for x, and y + yo for y, subtracting the original equation, and dividing by o, we get
3a⅜ - 2axx + ayx+ axÿ - 3y 2 ÿ + 3xx 2 o + &c.
Hence, regarding o as an evanescent quantity, we obtain, neglecting the powers of o,
3a⅛ - 2axx + ay± + axÿ - 3y 2 y = 0; consequently x ∙. y = 3y 2 - ax : 3≈ 2 - 2ax + ay.
This, as Newton observes, furnishes a ready method of drawing the tangent at any point on a curve. In fact, it is, changing the notation, equivalent to Barrow’s method already considered. Newton adds, that in like manner we may neglect, in all cases, the terms multiplied by the second and higher powers of o, and thus find an equation between x, y and their fluxions ⅛, y.
A good deal of confusion has arisen from the word fluxion having been commonly employed by the early English writers in the sense of an infinitely small increment. Thus, as is abundantly shown by Professor De Morgan in his tract on the early history of infinitesimals in England {Phil. Mag., 1852), all the early writers on fluxions, up to 1704, except Newton and Cheyne, employed the notation x to represent an infinitely small increment, calling it a fluxion. It is even remarkable that, in the extract from the Commercium Epistolicum which we have given, the words moment and fluxion seem to have been employed as synonymous. It should also be observed that in Newton’s earliest papers his method is strictly infinitesimal; and in the first edition of his Principia (1687) the description of fluxions is founded on infinitely small increments; so that the original conception of the calculus in England, as well as on the Continent, was based on infinitesimal principles.
Objection has frequently been made to Newton’s method of fluxions, that it introduced a foreign idea, namely, that of motion, into geometry and analysis. This objection is scarcely well founded, and was indeed answered by Newton when he stated that all his method contemplates is that one of the variables should increase uniformly (scquabili fluxu) as we conceive time to do.
Leibnitz, like Newton, supposed any variable magnitude as continually increasing or diminishing, by momentary increments or decrements. These instantaneous changes he regarded as infinitely small differences. Thus the infinitely small difference of a variable u was represented by du. His calculus also, like Newton’s, had two parts :—(1) the differential calculus, which investigated the rules for deducing the relation between these infinitely small differences of quantities from the relation which exists between the quantities themselves; (2) the integral calculus, which treated of the inverse problem, viz., the determination of the relation of the quantities when that of their differences is known. This corresponds to Newton’s inverse method of fluxions, as the differential calculus does to his direct method. It is not necessary to go into further detail hero on Leibnitz’s method, as it will be more fully considered subsequently; in fact, all our treatment of the calculus will be merely a development of this method.
The infinitesimal calculus had in the outset its opponents, such as the Abbé de Catelan, a zealous Cartesian, who declared in his Logistique Universelle, et Methode pour les Tangentes (1694), that it would be better to extend the principles of the Cartesian geometry than to seek for new methods; and this was said in the preface of a book composed on the principles, somewhat disguised, of the very calculus of which he was an opponent. It had another adversary in Nieuwentijt, a man who had written some tolerable works on morality and religion, but who had slight pretensions to be regarded as a geometer. Catelan was satisfactorily answered by De l’Hôpital, as was Nieuwentijt by Leibnitz, and afterwards by Bernoulli and Hermann, who proved that this adversary of the calculus really did not know what he opposed. For instance, Nieuwentijt, while admitting differentials of the first order, rejected all those of higher orders. For such a difference of treatment there is no foundation, for, if we imagine in a circle an infinitely small chord of the first order, the versine is an infinitely small line of the second order.
The calculus had a more formidable enemy in Rolle, a skilful algebraist, but a man full of confidence in his own notions, rash in forming his opinions, and jealous of the inventions of others. He attacked the certainty of its principles, and attempted to show that its conclusions were at variance with those obtained by methods previously known, which were acknowledged to be. correct. His attack was repelled by Varignon, who completely obviated the objections to the truth of the principles. These disputes occupied the French Academy a considerable part of the year 1701. The members were chiefly mathematicians advanced in years, who had been long accustomed to other methods, and were therefore not much disposed to receive new doctrines. Some took no part in the dispute, yet were not sorry to perceive a storm raised against a theory for which they had no great liking; others, more under the influence of their passions and prejudices, declared open war against it. Rolle brought forward objection upon objection; and, although Varignon answered them in succession, yet the former always claimed the victory. In the end the dispute degenerated into a quarrel, and commissioners were appointed to decide on it. These were Gouye, Cassini, and De la Hire. They, however, pronounced no judgment; but the public opinion, or at least the opinion of geometers, was in favour of Varignon. The first controversy thus ended, or rather was suspended for want of a decision from the commission; but Rolle soon renewed hostilities. The defence was next taken up by Saurin. The ground of attack was the indefinite form which the calculus gives for the subtangent of a curve at a point where two branches intersect each other, and which in this case is expressed by the fraction Saurin’s answer was satisfactory; but Rolle, intrenched in masses of calculation, obstinately maintained the combat. The Academy was again appealed to in 1705. The Abbé Bignon, who conducted its affairs, undertook to decide the controversy, with the assistance of Gallois and De la Hire, two judges by no means favourable to Saurin. They gave no absolute judgment, but recommended Rolle to conform more strictly to the rules of the Academy, and Saurin to forgive the proceedings of his adversary. Rolle afterwards did justice to the calculus by acknowledging his error in opposing it, and admitted that he had been urged forward by malevolent persons, one of whom was the Abbé Gallois.
Mathematicians have differed as to the best way of expounding the principles of the calculus. Newton, as has already been stated, employed the theory of motion as the means of connecting its doctrines with the principles of ordinary analysis. Leibnitz, again, with the same view, conceived quantity as passing from one degree of magnitude to another by the continual addition of infinitely small parts. The mind finds no great difficulty in distinctly apprehending the subject in either way. Objections have, however, been taken to both, and attempts made to substitute a better. Euler considered the infinitely small quantities of Leibnitz as absolutely zeros, that have to each other ratios derived from those of the vanishing quantities which they replace. D’Alembert proposed to make the basis of the calculus the consideration of the ratios of the limits of quantities. This method, as was indeed stated by D’Alembert, does not differ in any material respect from Newton’s prime and ultimate ratios. An English mathematician, Landen, substituted for the Newtonian method of fluxions another purely algebraical. His views are contained in a work entitled The Residual Analysis, a new branch of the Algebraic Art (1764). Lagrange, too, in the Memoirs of the Berlin Academy for 1772, proposed to base the calculus altogether on the expansion of functions, and thus to establish it on algebraical principles merely. He subsequently developed his method in his Théorie des Fonctions Analytiques lyl⅛T), and in his Leçons sur le Calcul des Fonctions (1806). Lagrange, however, adopted the infinitesimal method as the basis of his most important work, viz., the Mécanique Analytique. He states in his preface to its second edition (1811) that “when we have properly conceived the spirit of the infinitesimal method, and are convinced of the exactness of its results by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of abridging and simplifying our demonstrations.”
We shall close this introduction with a list of works on the subject.
Principal Works bearing on the Infinitesimal Method before the Invention of the Calculus.— Kepler, Nova Stereometric Doliorum Vinariorum, 1615; Cavalieri, Geometria Indivisibilium, 1635; Id., Exercitationes Geometricae Sex, 1647; Descartes, Géométrie, 1637; Torricelli, De Sphaera et Solidis Sphaeralibus, 1644; Grégoire St Vincent, De Quadratura Circuli, 1647; Huygens, Theoremata de Quadratura, 1647; Id., Horologium Oscillatorium, 1673; Waliis, Arithmetica Infinitorum, 1655; Id., Opera Mathematica, 3 vols., 1693-99; Fermat, Opera Varia Mathematica, 1679; Mercator, Logarithmotechnia, 1668; James Gregory, Vera Circuli et Hyperbolae Quadratura, 1668; Barrow, Lectiones Geometricae, 1670; Slusius, “Tangents to all Geometrical Curves,” Phil. Trans., 1672; Wren, “Rectification of the Cycloid,” Phil. Trans., 1673; Bullialdus, Arithmetica Infinitorum, 1682.
List of some of the Principal Works on the Calculus.— Newton, De Analysi per Aequationes numero terminorum infinitas, circulated in MS. inl669 (extracts from this memoir appeared in the 2d vol. of Wallis’s works, 1693, which comprehends the first publication to the world of the method of fluxions); Id., Principia, 1687; Id., Tractatus de Quadratura Curvarum, published with his Optics, 1704; Id., Methodus Differentiatis, 1711; Leibnitz, “NovaMethodus pro maximis et minimis, itemque tangentibus,” Acta Erud., 1684; Leibnitz et Bernoulli, Gommer. Epis. Ph. et Math., 1745; John Bernoulli, “Inventio Lineae Brachistochronae, ,, √4ciαA>w(Z., 1696; Id. Analysis Problematis Isoperimetrici, 1697; Id., Opera Omnia,17i2; James Bernoulli, Opera, 1744; De l’Hôpital, Analyse des infiniment Petits, 1696; Cheyne, Fluxionum Methodus Inversa, 1703; Hayes, Treatise on Fluxions, 1704; Manfredi, De Construe. Aequat. Diff. Primi Gradus, 1707; Taylor, Methodus Incrementorum, 1715; Stirling, Din. Tert. Ordin. Newtoni, 1717; Hermann, “De Construe. Equat. Diff.,” Comm. Petrop., 1726; Fontenelle, Élémens de la Géométrie de l’Infini, 1727; Clairaut, “Determinatio Curvae ejusdem Diff.,” Acta Erud., 1729; De Moivre, Miscellanea Analytica, 1730; Hodgson, Fluxions, 1736; Simpson, Fluxions, 1737; Maclaurin, Fluxions, 1742; Donna Agnesi, Instituzioni Analitiche, 1748; Euler, Meth, invcn. Lin. Cure. max. vet min. prop, gaud., 1744; Id., Introd. Analy. lnfin., 2 vols., 1748; Id., Institut. Cal. Diff., 2 vols., 1755; Id., Institut. Cal. Integ., 3 vols., 1768-70 (the titles of Euler’s numerous memoirs on the Differential and Integral Calculus are given in the edition of his Differential Calculus published at Pavia in 1787); Walmesly, Analyse des Mésures, des Rapports, et des Angles, 1750; Stirling, Methodus Differentiatis, 1753; Bougainville, Traité du Calcul Intégral, 1754; Landen, Mathematical Lucubrations, 1755; Id., Residual Analysis, 1764; Id., Mathematical Memoirs, 1780; Saunderson, Method iff Fluxions, 1756; Kästner, Separatio Indeterminat. in Aequat. Diff., 1756; D’Alembert, Opuscules Mathématiques, 1761-80; Robins, Mathematical Tracts, 1761; Waring, Miscellanea Analytica, 1762; Id., Meditationes Analyticae, 1776; Condorcet, Du Calcul Intégral, 1765; Le Seur et Jacquier, Elémens du Calcul Intégral, 1768; Lexell, “Methodus integrandi Aeq. Diff.,” Comm. Pctrop., 1769; Fontaine, Traité du Calcul Diff. et Intégral, 1770; Gianella, De Fluxionibus et eurum Usu, 1771; Cousin, Traité du Calcul Différentiel et Intégral, 1776; Laplace, “L’Usage du Calcul, aux Diff. part.,” Mém. de l’Acad., 1777; Condorcet, “De Integ. cujusdam Aequationis,” Comm. deBonon., 1783; Paoli, Memoriasull' equazione a differenze finite e parziali, 1784; Monge, “Sur le Cal. Int. des Equat. aux Diff. part.,” Λf<i∞. de VAcad., 1784; Charles, “Recherches sur le Calcul Intégral,” Mém. de l’Acad., 1784; L’Huillier, Exposition des Principes des Calculs Supérieurs, 1786; Id., Princip. Calculi Diff. et Integ., 1795; Mascheroni, Annotationes ad Cal. Integ. Euleri, 1790; Tabiescen, Principia atque Historia Calculi Diff. et Integ. neenon Methodi Fluxionum, 1793; Lagrange, “Calcul des Variations,” Mise. Taur., vols. ii. and iv., 1760-69; Id., Théorie des Fonctions Analytiques, 1797; Id., Leçons sur le Calcul des Fonctions, 2d ed., 1806; Id., separate Memoirs, edited under the care of Serret, 7 vols., 1867-77 (the remainder of his works are in course of republication in the same series); Vince, Principles of Fluxions, 1797; Carnot, Réflexions sur la Métaphysique du Calcul Infinitésimal, 1797; Lacroix, Traité du Calcul Différentiel et du Calcul Intégral, 1797; Arbogast, Calcul des Dérivations, 1800; Legendre, Exercices de Calcul Intégral, 3 vols., 1811-19; Id., Traité des Fonctions Elliptiques, 3 vols., 1825-28; Cauchy, Cours d’Analyse, 1821; Id., Appl. Géom. du Cal. Infin., 1823; Id., Mém. sur les int. déf. prises entre des limites imag., 1825; Id., Leçons mir le Calcul Différentiel, 1829; Ohm, Μ., System der Mathematik, 9 vols., 1822-52; Id., Lehrbuch f. d. gesummte Höh. Math., 2 vols., 1839; Magnus, Sammlung von Aufgaben d. Analyt. Geom., 1833; Navier, Leçons d' Analyse de l'Éc. Polyt., 1840; Moigno, Leçons de Cal. Diff. et de Cal. Inh, 2 vols., 1840-44; Id., Calcul des Variations, 1861; Duhamel, Cours d ’ Analyse de ÎÉc. Polyt., 2 vols., 1840-41; 3d ed. by Bertrand, 2 vols., 1874-75; Cournot, Théorie des Fonctions et du Calcul Infinitésimal, 1841; Gregory, Examples on the Diff. and Int. Calculus, 1841; De Morgan, Differential and Integral Calculus, 1842; Hymers, Integral Calculus, 1844; Schlömilch, Handbuch der Differenzial-und Integralrechnung, 1847; Id., Compendium der Höheren Analysis, 2 vols., 1874; Minding, Sammlung von Integraltafeln, 1849; Meyer, Exposé Élém. de la Théorie des Int. Déf, 1851; Todhunter, Differential and Integral Calculus, 2 vols., 1852; Id., On Functions of Laplace, Lamé, and Bessel, 1875; Price, Infinitesimal Calculus, 2 vols., 1854; Bierens De Haan, Tables d'intégrales définies, 1858; Id., Exposé de la théorie des intégrales définies, 1862; Boole, Differential Equations, 1859; Id., Calculus of Finite Differences, 1860; Grassmann, Die Ausdehnungslehre, 1862; Bertrand, Traité de Cal. Diff. et de Cah Int., 2 vols., 1864-70; Meyer, G. F., Vorlcs. Ü. d. Theorie d. bestimmten Integrale, 1871; Williamson, Differential and Integral Calculus, 1872-74; Hermite, Cours d’Analyse, 1873; Durége, Theorie d. Funktionen einer complexen verändert. Grösse, 2d ed., 1873; Folkierski, Principles of Diff. and Inh Calc. (Polish), Paris, 2 vols., 1873; Rubini, Elementi di Calcolo infinitesimale, 2 vols., 1874-75; Serret, Cours de Cale. Diff. et Inh, 2d ed., 2 vols., 1878-79 (the 8th edition of Lacroix’s Traité Élémentaire, by Serret and Hermite, contains in the notes many valuable additions); Riemann, Gesam. Math. Werke, 2d ed., 1876; Id., Partielle Differentialgleichungen, 2d ed., 1876; Lipschitz, Lehrbuch der Analysis, 2 vols., 1877-80; Hoiiel, Cours de Calcul Infinitésimal, 3 vols., 1878-79; Boucharlat, Él. de Cale. Diff. et Int., 8th ed. by Laurent, 1879; Stegemann, Differential-und Integralrechnung, 2 vols., 3d ed., 1880.
The preceding list contains the names of some of the most important existing treatises on the calculus. It makes no pretence to completeness; in fact, many of the most valuable contributions to the subject are published in the numerous mathematical journals, and in the transactions of learned societies. In treating of elliptic and hyperelliptic functions we shall give a short list of the chief works on that great branch of the calculus.
PART I.
Differential Calculus.
1. In the application of algebra to the theory of curves and surfaces some of the quantities under consideration are conceived as having always the same magnitude, such as the radius of a given circle or of a given sphere, or the axes of a given ellipse or hyperbola; others again are indefinite, and may have any number of particular values, such as the coordinates of any point on a curve. This difference naturally suggests the division of the quantities involved in any question into two kinds, one called constants, the other variables.
It is usual in analysis to denote constants by the first letters of the alphabet, a, b, c, &c.; variables by the last, u, v, w, x, y, z, &c.
2. One quantity is said to be a function of another when they arc so related that any change made in the one causes a corresponding variation in the other. This relation may subsist whether there exist an expression for the function by which its value is determined for each value of its argument ; or the relation may sometimes be defined by certain characteristics of continuity and discontinuity. When an expression is presupposed the relation is usually represented by the letters F, f, φ, &c. Thus the equations
∙u=F(x), v∙=flx), w=φlx), denote that u, v, w are regarded as functions of x, whose values are determined for any particular value of x when the forms of the functions are known.
In each of these expressions the argument x is regarded as the independent variable, to which any value may be assigned at pleasure; and u, υ, w are called dependent variables, as their values depend on that of x, and are determined when it is known.
For example, in each of the equations y = 10 x , y = tan x, y=fLf-, (L — ∙ZJ
the value of y is known when that of x is given.
Such functions are called explicit. ’
3. In many cases a variable y, instead of being given explicitly in terms of x, is connected with it by an equation of a more complicated character. For instance, suppose them connected by the relations x log y =y logae, sin y ≈x sin la+f), y 3 +aP + 3axy = 0; in these cases the value or values of y may be supposed known when x is given, and y is said to be an implicit function of x. Such cases are comprehended in the form
≠(α,y) = 0 .
1 n such a form y may be regarded as an implicit function of x, or x as an implicit function of y, at pleasure.
4. Again a quantity may be a function of two or more independent variables. Thus in the equation m =sin lax + by), x and y may be regarded as independent variables, and u as a function of them. Such functions are in general denoted by Φlx, y), φlx, y, z), &c.
5. A function φlx) is said to be continuous between any limiting values of x, such as a and b, when to each value of x between those limits there corresponds a finite value of the function, and when an indefinitely small change in the value of x produces only an indefinitely small change in the function. In such cases the function in its passage from any one value to any other between the limits receives every intermediate value, and does not become infinite. This continuity can be readily illustrated by taking φlx) as the ordinate of a curve, whose equation may then be written y = ≠(x).
6. If the variable x be supposed to receive any change, such change is called an increment ; this increment of x is usually represented by the notation A®. A decrement is regarded as a negative increment. When the increment, or difference, is supposed to be indefinitely small, it is called a differential, and is represented by dx ; i. e., an infinitely small difference is called a differential.
In like manner if « be a function of x, and x become x + ∆x, the corresponding value of u is denoted by u + ∆u; i.e., the increment of u is represented by ∆u. For finite increments of x it is obvious that the ratio of the increment of u to the corresponding increment of x has, in general, a finite value. Also when the increment of x is regarded as being indefinitely small we find that the above mentioned ratio, i.e., ~ , has in general in each case a definite limiting value; and the first study of the differential calculus necessarily involves the investigation of such limiting ratios for the different forms of functions of x.
In fact we have seen that the differential calculus took its rise from the investigation of the limiting value of the ratio of the increment of the ordinate y to that of the abscissa x, so as to find the position of the tangent at any point on a curve.
Thus if the equation of a curve, referred to rectangular axes be denoted by flx,y) = 0, then ⅛, i.e., the limiting value of for any dx ∆x j point on the curve, represents the tangent of the angle which the tangent at the point makes with the axis of x.
7. Again, if we suppose x to become x + h (where h represents ∆x, the increment of«) in the equation u≈flx), then the increment of u is represented by flx + h)-flx), and — = ∕(ic +⅞)-.Λ¾) hence du h
— represents the limit to which flx + h)-flx) h approaches indefinitely, when h is diminished without limit.
There are two methods in general of finding this value of — .
da;
The first consists in determining the limiting value of∙ ^ c + ~Λ x ) h by decreasing h indefinitely. The second consists in expanding f(x + h) in a series of ascending powers of h, and taking the coefficient of h in the expansion. This is the method introduced by Lagrange when he proposed to make the calculus a branch of ordinary algebra, and altogether independent of the consideration of infinitely small magnitudes, or of limits.
It is easily seen, as was shown by Lagrange, that the result obtained by the latter method is the same as that arrived at by the former; for, since flx + h) becomes flx) when Λ = 0, ∕(z)is the first term in the expansion, and we may assume f lx + h) ≈flx) +ph + qh 2 + &c., in which p, q, &c., represent functions of x, independent of h, then ∕⅛÷*>-∕w. f , 4 . i ⅜ +tc .
If now we suppose h = 0, the left hand side reduces to p ; and, accordingly the coefficient of h in the expansion of∕α + A) is the limiting value of the expression ∕C c ÷^ , ) ~∕(≈) . h
This coefficient of h was called by Lagrange the first derived function of the original function/;®), and he represented it by the notation flx).
Hence we have dx dx
In this cs.se f'lx)dx is called the differential of/®), and flx) is called its differential coefficient.
8. We have already seen that the principles of the calculus may be regarded either from the consideration of limits, or from that of infinitesimals or differentials; the former was the method adopted by Newton, in his later investigations at least; the latter was that adopted by Leibnitz.
The limit of a variable magnitude may be defined as follows. If a variable magnitude tends continually to equality with a certain fixed magnitude, and approaches nearer to it than any assignable difference, however small, this fixed magnitude is called the limit of the variable magnitude.
For example, if we suppose a polygon inscribed in or circumscribed to any closed curve, and afterwards imagine each side indefinitely diminished, then the closed curve is said to be the limit of either polygon. By this means the whole length of the curve is the limit of the perimeter of either polygon, and the area of the curve is the limit to the area of either polygon.
9. The following principles concerning limits are of frequent application. (1) The limit of the product of two quantities, which vary together, is the product of their limits. (2) The limit of the quotient of two quantities is the quotient of their limits. These are nearly self-evident propositions; they may, however, be formally proved as follows.
Let P, Q represent the variable quantities, and p, q their limits; then, if P= j p + α, and Q=γ + 3> «, ß denote quantities which diminish indefinitely as P and Q approach their limits, and become evanescent in the limit.
Again, TQ,=pq+pβ + qa +aβ
Accordingly in the limit, PQ =pq.
The corresponding theorem for the quotients is established easily in like manner.
10. Again, if we conceive any finite number or magnitude to be divided into a very great number of equal parts, each part is very small in comparison with the original magnitude. By supposing the number of parts to be increased indefinitely, i.e., so as to exceed any assigned number, however great, then each part may be regarded as indefinitely small in comparison with the proposed magnitude, and may be called an infinitesimal with regard to it.
By an infinitesimal, or an indefinitely small magnitude, we understand a magnitude which is less than any assigned magnitude however small, and which can be diminished indefinitely, so as to approach as near as we please to zero, without ever absolutely attaining to it. For instance, the difference between the area of a circle and that of an inscribed regular polygon can, by increasing the number of sides of the polygon, be made less than any assigned area, however small; but, no matter how large the number of sides may be, this difference can never become absolutely zero. It would be easy to give other illustrations of the sense in which the word infinitesimal is employed in analysis.
11. Again, if a be regarded as an infinitesimal of the first order, α 2 , being infinitely small in comparison with α, is regarded as an infinitesimal of the second order. In like manner α 3 , a 4 , . . . a n , may be regarded as representing infinitesimals of the third, fourth . . . nth orders, respectively.
Again, two infinitesimals a, β are said to be of the same order if the fraction — tends to a finite limit. If — tends to a finite limit, a a n β is called an infinitesimal of the nth order in comparison with a.
12. To avoid misconception, it should be borne in mind that infinitesimals are not regarded as being actual quantities in the ordinary acceptation of the words, or as capable of exact representation. They are introduced for the purpose of abridgment and simplification of our reasonings, and are an ultimate phase of magnitude when it is conceived by the mind as capable of diminution below any assigned quantity, however small. Such magnitudes are in all cases, as styled by Carnot, auxiliary quantities, introduced for the purpose of facilitating our investigations, but they should disappear from our final results.
We shall illustrate this statement by the example of drawing a tangent to a curve,—in which problem the method of infinitesimals may be said to have originated. We introduce the infinitesimals dx and dy, for the purpose of finding their ultimate ratio, i.e., in order to determine the limit of Now this limit is in all cases dx a function of x and y, the coordinates of the point of contact, and cannot contain in it either dx or dy, since they must be taken as evanescent quantities when we proceed to the limit.
Likewise in all other applications of infinitesimals in the differential calculus, we endeavour to find the ultimate ratio of two indefinitely small quantities, or infinitesimals; and it is unnecessary to attach any precise meaning to such infinitesimals during the course of our investigations, further than to regard them as variable quantities, which become evanescent when we proceed to our final results.
In employing infinitesimals in such cases, we proceed on the principle that the limit of the ratio of two infinitesimals, a and β, is the same as that of α' and β', provided the limit of — ≡ 1, and limit α
β ’
0f77-=l. This is evident since, in all cases, we have P a _ a a' β' * ~β~~fi · β' ’ ~β' , a result which must hold in the limit.
In consequence of this principle, before proceeding to the limit, we may neglect an infinitesimal of any order in comparison with one of a lower order. For instance, in seeking the ultimate ratio of γ to δ, where γ = A 1 α + A 2 a 2 + A 3 a 3 + &c. + A n a n + &c.,
δ = B 1 3 + B 2 3 2 + B 3 0 3 + &c. + B n 0 n + &c., in which A 1 , A 2 , A 3 . . . B 1 , B 2 , &c., are finite, and independent of the infinitesimals a and β, we may neglect α 2 , a 3 , 3 2 , β 3 . . . in comparison with a and β, and we get the limit of—× limit of — .
δ B 1 β
In general, if a, β be infinitesimals of the same order, their ratio has a finite magnitude; and if 7=∕ 1 (ct, β'), δ=√ a (α,3), then in finding the limiting value of X. we take the terms of the lowest δ
order in a and β in f 1 and ∕ 2 , neglecting all infinitesimals oi higher orders; substituting in the result the limiting value of — , β we obtain the required limit for-2-.
δ
13. Again, if α 1 + α 2 . . . +a n represents the sum of a number of infinitely small quantities which approaches a finite limit when n is indefinitely increased, and if β 1 , β. 2 . . . β n , be another system of infinitely small quantities, such that
⅛- = l+.e 1 , — = l + e 2 , . . — = l + ∈ n
«1 α a a n where e 1 , e 2 . . . e n become evanescent in the limit, then, when n is indefinitely increased, the limit of the sum of 0 1 , β. 2 . . . β n is equal to that of a 1 , α 2 . . . a n . This is evident from the elementary algebraic principle that the ratio 0ι + ⅜ r ♦ · + & p e3 between the greatest and
«1 + «3 · .. + ‰
the least values of the fractions ⅛-, ⅛, - ι t êï; and it accordingly c⅛ι α 2 a∏
has unity for its limit, under the supposed conditions. For example, suppose any magnitude divided into a number of parts, and that each is capable of subdivision into two parts, one of which can be simply found, and the other not so. Let A + a be the first part, of which A is of the former species, and a of the latter. In like manner let B + b, C + c, &c., be the other parts. Then the required magnitude is represented by A + B + C+ &c. +α + ⅛÷c + &c. Now suppose that when a sufficiently great number of parts is taken we can make a, b, c, &c., as small as we please in comparison with A, B, C, &c., then α + δ + c+ &c., can be made as small as we please with respect to A + B + C + &c.; consequently by continuing the process indefinitely, the limit of the sum of A + B + C + &c., is equal to the required magnitude, without the necessity of paying any attention to the remaining parts. This latter may be regarded as the fundamental principle of the integral calculus, and the former, given in § 12, as that of the differential.
14. In consequence of metaphysical objections to the employment of infinitesimals, many writers on the calculus have confined themselves exclusively to the method of limits or limiting ratios, and by so doing have in many cases involved themselves in long and cumbrous demonstrations of theorems which follow with great facility by the adoption of infinitesimals. In reality the difference between the method of infinitesimals and that of limits (when exclusively adopted) is, that in the latter method it is usual to retain evanescent quantities of higher orders until the end of the calculation, and then to neglect them. On the other hand, such quantities are neglected from the commencement in the infinitesimal method, from the conviction that they cannot affect the final result, as they must necessarily disappear when we proceed to the limit. A very little reflexion will show that the result obtained in both cases must be the same. Moreover such quantities are neglected, not, as Leibnitz stated, because they are infinitely small in comparison with those that are retained, which would produce an infinitely small error, but because they must be neglected to obtain a rigorous result; since such result must be definite and determinate, and consequently independent of these variable indefinitely small quantities. It may be added that the precise principles of the infinitesimal calculus, like those of any other science, cannot be thoroughly apprehended except by those who have already studied the science, and made some progress in the application of its principles.
15. The preceding statements may also be regarded in connexion with the different meanings of the terms “zero” and “an evanescent quantity.” There is but one process in arithmetic which yields an absolute zero, namely subtraction, thus α-α = 0. But from no other arithmetical process does zero arise, except by the same train of ideas as leads us to the use of the word infinite. Wβ cannot, for example, obtain the quotient zero by dividing one finite magnitude by another. We can make the result as small as we please, but not absolutely zero. When, therefore, we consider an equation made by addition or subtraction of terms, the absolute zero may be used without reservation, thus c λx + a = b, and 2x + α-δ=0 may be substituted for each other without any particular examination of the symbol 0. But in any other case we consider zero or 0 as the limit towards which we approach by a series of diminutions, none of which is final. Thus when we see that we can nearly arrive at a certain conclusion by attributing a small value to a particular magnitude, that wo can more nearly attain this conclusion by attributing to it a smaller value, and so on without limit, i.e., that we can approximate to this conclusion as nearly as we please by the use of a value as small as we please, but that we never attain it as long as the magnitude has any finite amount, then such conclusion is said, for abbreviation, to be absolutely true when the magnitude is nothing or zero. These considerations will help to explain the sense in which Euler was correct when he stated that dx and dy in the calculus must be regarded each as zero.
16. We now return to the consideration of the method of finding the derived functions or differential coefficients of the different forms of functions of x. Before doing so, hoλvever, it will be necessary to establish two or three general principles.
We commence with the differentiation of a product.
Let y≈uv, where u and ν are functions of»; then
∆y= (u + ∆a)(t> + ∆v) - uv =u∆ν + (y+∆υ)∆u;
∆w ∆ν . . ∆tt λ ^Γ~ + (^ + Δr) — 5
∆x ∆x κ , ∆x proceeding to the limit, this becomes dluv∖ dv du -⅛- = u ^T^ +v -T- ' dx dx dx
In like manner, if y = uvw, we got du du dv , dw , — = vw - i - + wu - r - + uν-y- j dx dx dx dx and, in general, if y≈y,y i y 3 · · y» we have
1 dy = f_ ⅜ι l 1 ⅛ + + JL ⅜"
y dx y l dx y. i dx ' ' ' y n dx
Again, if y=-^-, wehavew≈vy, consequently du dy dv j-≈v -r + y^-; dx dx dx du dv dy 1 du u_ dv dx dx dx v dx v 2 dx v 2
17. Next, to differentiate a function of a function of x ; let y= f(x), and u=<∕>(y), to find∣^. Suppose y 1 , u i to bo the values which y and u assume when x becomes x 1 , then u 1 -u u 1 -u y,-y ∆u-∆u ∆y x 1 - x y 1 -y ’ «i - « ’ ∆χ ∆y ∆χ'
Hence, since, as proved already, the limit of the product of two variable quantities is the product of their limits, we have du dïi dy dx~ dy' dx
Consequently, the derived function of u with respect to x is the product of its derived with respect to y and of the derived of y with respect to x. Again, if we suppose u≈x, our equations become y=f(x), and x≈φ(y). In the former y is regarded as a function of x, and in the latter x as the corresponding function of y.
Such functions are said to be inverse to each other; and in this case we have—
. dx dy dx _ dy l=>- 7 - · -f > or ^~ = 1÷≠ · dy dx dy dx
18. There exist in analysis a small number of simple or elementary functions, each of which requires a special investigation in order to find the corresponding derived function. When these have bδen established the differentiation of functions composed of these elementary functions can be readily obtained, by applying one or more of the principles just established.
19. We commence with the equation y=x n , in which n is a constant.
(1) Let n be an integer, and y 1 the value which y assumes when x becomes x 1 ; then
Z lξ j∕^∙≤ 1+ n . a+ . ι i +ajn . 1 .
aq- x x l -x i
Now the limit of the right hand side when ae=aq is nx n ~ 1 ; accordingly we have in this case dx n „ 1
-—=τιa 3 n- 1 .
dx m.
(2) Let y≈x n , where m and n are integers. Here y n =x m , and accordingly nw"^ 1 ⅛=in≈"^ 1 ; hence we get ⅛ = - χ " 1 . o j dx dx n
(3) Let y=x ”*=—; then, from § 16, we get ⅛ = -mx m ∖ • x m dx
Consequently we get the following rule, applicable in all cases, for the differentiation of a power of x :— Diminish the index by unity, and multiply the power of x thus obtained by the original index.
20. We shall next consider the elementary circular and trigonometrical functions.
Let y = sin x. Then y 1 =sin(« + h); y,-y sin («+Ä)-sin x 2 . h f , h ∖ -h--^~lτ y≡-2 cos (ae + y;·
β u t -Ê. s in — becomes unity in the limit, and consequently h 2
⅜
-f- = cos X. dx
In like manner it is easily seen that
⅛ 0 lΞ=- 8i na=.
dx d sin x . d cos x
Ao·ain »tan« _ d sin x dx dx ’ dx dx I cos x I - cos 2 «
cos¾ + sin 2 « 1 .
E≡⅛- -≡5- wλ ≈∙
Similarly ⅛ L-,———∞≈t∞≈
dx sιn¾ dx
Corresponding to these trigonometrical functions we have the circular functions, sin* 1 «, cos* 1 «, tan* 1 «, &c.
If y = sin*⅛, we have « = sin y, and hence ^-oosy,or⅛- 1 ’ .
dy dx coay ^√l-x 2 d sin -1 « 1 d x √ι~χ 2 '
In like manner d cos* 1 « _ - 1 d tan* 1 « _ 1
d« √j - x 2 ’ dx ~ 1+« 2 '
21. Next, let ι∕=log α z. Here y 1 -> log α (z÷λ);
∙∙∙ V4 1 √⅛ a )-W 1 ÷⅜)
Let — = ïi ; then x
⅛ 1 °ε∙(1 ÷⅜)-⅛ t 1 °8- (ΐ+·*)-|ΐοβ.(ΐ+«Λ· fl∕ ∖ db J dblb <h
The limiting value of (1 + u} " when u * 0, i. e. , of ι q. — when « increases indefinitely, is represented by the letter e (see Algebra, vol. i. p. 558), and is the base of the natural or Naperian system of logarithms. Hence we have
⅛.1⅛,. dx x
If e be taken as the base of our system of logarithms, we have d log x 1 dx x
In our subsequent investigations we shall suppose all logarithms, unless otherwise specified, referred to this base, and omit the suffix.
22. The method of differentiation of an exponential function follows immediately from the preceding.
For let y≈a x , then log y≈x log a,
.∙. JL ⅛ = log a , or log a . y dx dx
We add a few examples for the purpose of showing the application of the preceding results to the differentiation of more complex functions.
(1) ι∕=x*.
Here log y=*x log x; . , . — -^ = log« + l.
y dx
Hence ⅛ = (1 + log x)x x . dx
(2) y = log -*≈
√α 8 +ic a
Here y ≈ log x - ⅛ log (a s + x 2 );
1 x a* dx°~ x a i +x 2 ^x{a 2 +x 2 ) '
(3 ) y =log √⅛t√S
√l+iB-√I-a!
tr 11 ∕Vl + a! + √1 -aλ 2 1 . l + Vl-ae 1 Here w=⅜ log! , .-. r.) = ⅜ log 7 -
j 2 Λ√l + a≈-√l-α√ “l-Vl-ae 3
=⅛ log (1+ ∖∣∖ '~χ 2 '} - ⅛ 1 °g (1 - √ι-⅛≡); . dy -x x d χ ~ 2√lWic 2 (1 + √Γ l T 2 ) ~ 2√Γ^ 2 (1 -
-1
(4) Prove that sin¾-^ (sin"« sin nx)=n sin n +⅛ sin (n + 1)«.
dx
Here — (sin”« sinn«) = n sin” - ⅛ (cos « sin nx + sin « cos nx) dx = n sin" -1 « sin (n + 1)«; . ’. &c.
. _i √l+∞ i + √l-ic il w > , - tan √≡.√π-
Here ^ 1 + ^ + ~≈ 8 -^tan y ; from this we get √l + κ 2 -√l -x 2
« 2 = sin2y; dy_ x x dx~ cos 2y √1 -χ 4 ’
(6) If y =log sin «, prove that = cot «.
(LX (7) If y=tf c , prove that -J- ≈ e χt χ x (l + log x). dx
(8) If y =—, prove that X dy + da;
Vl + y 4 λ∕l+α 4
(9 ) i⅛-ι 0g (√^+∖^).n≡>ttatg- 2 - 7 =-
23. We shall conclude this section with the consideration of the differential of the area ABPM (fig. 3) of a plane curve, comprised between the curve, the axis of x, and two ordinates, of which one BA is fixed and the other PM is variable, x, y representing the coordinates of P. This area, when the equation of the curve is given, is an implicit function of x. If it be represented by u, we proceed to find its differential coefficient, or
—. Suppose x to receive an in-iZ*i√
definitely small increment represented by MM', the corresponding increment of the area is represented by PMM'P', i.e., by the sum of the rectangle PMM'R and the elementary area PP'R. Now the latter area becomes evanescent in the limit in comparison with PMM'R.
Consequently in proceeding to the limit we have—-=PM = y, (JX or ⅛ =φ (x), where y = φ(x) is the equation of the curve.
dx
From this we can make an important inference, viz., that in all cases there exists a function whose differential coefficient is any given function of x, suppose φ(x). To find such a function it is sufficient to consider the curve whose equation in rectangular coordinates is y≈φ(x) ; then the area comprised between any fixed ordinate and the ordinate whose abscissa is ae is a determinate function,—which, by the preceding, has φ(x) for its derived function.
Successive Differentiation.
24. We have seen that from any function of a variable we can obtain by differentiation a new function, called its differential coefficient, or, after Lagrange, its derived function.
If the primitive function be represented by f(x), then, as already stated, its first derived function is denoted by f(x). If this new function, f(x), be treated in the same manner, its derived function is called the second derived of the original function /(ae), and is denoted by ∕"(,r). In like manner, the derived function of f , '(x) is the third derived oιf(x), and represented by f"(x), &c. In accordance with this notation, the successive derived functions of fix) are represented by f ∖ x),f ∖ x),H ∖ x), . . .f^Kx), each of which is the derived function of the preceding.
25. In like manner, if y≈f{x), then⅛=∕ , (ar) .
dx d J⅛f ∖
Hence dx dx j ' ' d (^ ∖
The function — y aι ∕ ⅛ written , and is called the second dx dx 1 differential coefficient of y with regard to x. d ^jL
Likewise - is written , and so on; dx dx i and the series of functions dy (By dfy d*y dx ’ dx i, dx i ' ' ’ dx* are called the first, second, third, . . . nth differential coefficients of the function represented by y.
It is sometimes convenient to adopt a notation analogous to that of fluxions, and to represent the senes of differential coefficients of y by y f ,y",f", f n ∖ in order to abbreviate the labour of writing down the system of successive differential coefficients.
26. It is plain that the determination of the series of successive derived functions of any function of x docs not require any new principles, as it is accomplished by successive applications of the methods already considered.
For example, if y=x n , we have dy ,=nx n ~ 1 , dx hence ^^=n(n- l)≈ n ^ 2 , ⅛(=n(n-l)(7i-2)rε n * 3 , &c.
Again, if y≈e ax , we have ft=ae ax , *⅛≈a 2 e ax , &c., dx dx∙ i and in general ^~-=a n e ax . dx n
27. We next proceed to a fundamental theorem due to Leibnitz, and first published in Mis. Βeτol., 1710, viz., to find the nth derived function of the product of two functions.
Let y=ιιυ ; then, if we write if, ιι', √, y", u", &c., for dy du dν d 2 y Tx λ dx’ dx ’ dB’ &C ·’ we have f—ud +vιι'.
The next differentiation gives ιf ’ =ιιd' + u'd + du' + vu" =u υ" + 2u'd + vu".
The third differentiation gives if" = ιιd" + u'v" + 2ιι ’ d , + 2u"d + du" + vu'" = nd" + Zν!d' + 3u"√ + vιi", in which the coefficients are the same as those in the expansion of (« + bf.
Suppose that the same law holds for the nth differential coefficient, and that y(n)= u dn) + nu ' ν (n- 1) ψ ⅛⅜~ 1 ) u " t (n - 2 ) + & c . j
1 . 2
+ nu< n - ¾' + u<'⅛; then, differentiating again, we get y(n + 1 ) = udn+l) -f- u , d n ) + n(u ’ d n ) + u!'ν Gι - 1 )
+ AZL≤] (√M n -i) + √"√n-a)) + fc c . . . . +n(n+⅜
= uri"+i) + (n +1 )√ri") + < 7ι ÷ 1 )rc w "√n -1)+&c. . . . ,
1.2
in which the coefficients follow the law of the Binomial Expansion.
Accordingly, if this law hold for any integer value of n, it holds for the next higher integer; but it holds when n∙=3, therefore it holds for n = 4, &c.
In the ordinary notation the preceding result is written d n (ud d n v du(B~ l ν n(n-l) cPu d a ~ 2 υ d n u dx n u dx n n dx dx n ~ 1+ 1.2 dB dx n ~ 2 ^ c ' ^'°dx^'
(1) If y=c αx sin bx, to find .
dx n
Here i -f=e ax (a sin bx + b cos bx). dx
Now let b=a tan φ, and we have
⅛≈(α 2 + δ 2 )i e ax (sin bx cos ≠ + cos bx sin φ) dx
= (α 2 + δ 2 )i e ax sin (bx+φ).
Similarly we get -dl =(α 2 + δ 2 ) e ax sin (bx + 2φ); cl n ιι n and, in general, = (a 2 + δ 2 )a c ax sin (bx+nφ).
(2) If y=cot^⅛, to find .
v , j ’ dx n
Hero * ≈ = cotw, .∙. -sin 2 y, dx hence A∣ (sin 2 y) = _ ⅛ . A (sin 2 y)
dx 2 dx dx dy = sin 2 y (sin 2 y)=sin 2 y sin2y . dy
Again (si∏ 2 y sin2y)=^ ~ (sin 2 y sin2y)
dx i dx dx dy
= - sin 2 w A (sin 2 y sin2y) dy
= - 1 . 2 sin s y sin3y. (Ex. 4, § 22. )
In like manner, t ~ i = ^∖- .2.3 sin 4 y sin 4y.
dB
An<l, in general, ( - l) n . |»-1 sin n y sinny.
(3) If y = z n ^ 1 logz, prove that .
dx n X
(4) If y=cos (α sin~ 1 z), prove that
(l-≈≈)⅛- a ,⅜ +α ⅛- 0 .
dx i dx a
(5) If y=a cos nx+b sin nx, prove that d 2 y 1 2 n ^ + 7i-y=0.
(6) If u≈xy, prove that ^y +n dn ' 1 y dx n dx n+ dx"-ι ’ Partial Differentiation.
28. We have hitherto treated of functions of a single independent variable solely. The principles established so far apply equally to the case of functions of two or more independent variables.
For example, in the equation u = ax 2 + 2bxy + cy 2 , the variables x and y may be capable of change independently of each other; and if we suppose x to vary, y remaining constant, the corresponding differential coefficient of u is represented by du , ,
— , and we have dx ~≈∙2ax+2by . dx
In the same case if we suppose y to vary, x being unchanged, the corresponding differential coefficient is represented by — , and we dy have ~ = 2δz + 2cy.
dy
In general, if «bea function of two variables, x and y, represented by the equation
74=∙≠(z, y), we have two differential coefficients
<⅛ θ r dφ{x , y} and du θ r dφ(x , y) . dx dx ’ dy dy '
These are called the partial differential coefficients of the function, with regard to x and y respectively. They are usually written ~and⅛ , and are plainly determined in the same manner as in dx dy the case of a single variable.
29. These new functions and admit of being treated in like dx dy manner. Thus the partial differential coefficient of ⅛, taken with dx ,dιι d ~d∙ respect to z, y being supposed unchanged, is represented by—-— dx or by⅛; likewise its differential coefficient with respect to y is (LX> represented by; and so on. dydx
30. It can be seen without difficulty that
, dw du d 2 u <Pu . dx dy dxdy dydx ’ ' dy dx ^∆u
In fact signifies the limit to which -~- approaches as ∆z and Ay diminish beyond limit.
Again ≠(z + ∆z , y)-≠(z , y)
Az Az
In like manner, δ Aw ≠(z+∆z, y + ∆y)-≠(z+∆z,y) _ ≠(z , y + ∆y)-≠(z , y) ∆x ∆x Az
∆y Ay —————
≠(z + ∆z , y÷∆y)-≠(z÷∆z, f)-φ(x, y + ∆ιf)+φ(x, y) ∆x∆y „ ∆u Δ
It is easily seen that - has the same value. Accordingly the limits of the two expressions must be equal, and hence we infer d 2 u _ d?u dydx dxdy '
31. In general, if w be a function of several independent variables z n z 2 , . . . z n , we obtain n partial differential coefficients of the first order, denoted by du du du du dx 1 ’ dz 2 ’ dx 2 ’ · ’ dx n ‘
In like manner, the partial differential coefficients of the second order are represented by d?u <Pu d 2 u , i
- 7 - s , -. , , , &c., and so on.
dx 1 i dx 1 dx 2 dx, i -
We have, as in the former case, between each pair of variables d?u d?u .
— &0
dx 1 dx 2 dx 2 dx l ’
32. In the equation u≈φ(x,y ∖ if we consider x and y to increase simultaneously, then, if ∆u represents the total increment of u, we have ∆u=φ(χ+∆x , y + ∆y)-φ(x , y)
≈φ{x + ∆x, y + ∆y)-φ(x, y + ∆y) + ≠(z, y÷∆y)-φ(z, y)
⅜>(z + ∆z,y + ∆y) -φ{x,y + ∆y) i φ(x, y + ∆y) - φ(x , y) _
Az ∆y y
If now we suppose Az and Ay to diminish indefinitely, and represent the corresponding differentials by du, dx, dy, we have in the limit , du, du , du=- r -dx+- dy. dx dy j
This is called the total differential of u, and it is readily seen that it is equal to the sum of the partial differentials arising from the separate increments in z and y. The same principle plainly holds in a function of any number of variables.
33. If u=φ (-ι>, w), where v and w are both functions of z, then by the preceding it is readily seen that du_du dν dw dw . dx dυ dx dw dx ’ and similarly for any number of functions.
34. The principles of total and partial differentiation admit of simple illustration in plane and in spherical trigonometry. For, in either a plane or a spherical triangle, we may regard any three - of the parts a, b, c, A, B, C as being independent variables, and each of the others as a function of the three so chosen.
For instance, in a plane triangle, if the sides a and b and the contained angle C be taken as the independent variables, we have c 2 = a? + δ 2 - 2ab cos C; v „„ de a - b cos (J „
hence — = = cos B:
da c likewise ⅛=cos A , ⅛ = α sin B, db ’ dG .. de =cos Bda + cos A db + a sin BdC.
dA.
Again, to find — , we have b sin Λ = α sin B = α sin (A + C); d ∖ j hence, regarding a and b as constant, we have
δ cos A ⅛5==α cos (A + C) fl + = - a cos B-α cos B; dG ' ∖ dC J dC ’ , dA_ acosB
‘ ' dC c ‘
In like manner we have, in the same case, dA sin B dA sin A ... sin B , sin A ,, a cos B
— =≈- , - 5r ∙= .∙. √A≈. da db dC. da e do e c e e
Again, in a spherical triangle, cos c=cos a cos δ + sin a sin b cos C.
From this we obtain
⅛ = cos B , ⅛ = cos A , ⅛ i = sin a sin B , da db dG
.∙. dc = cosB<fα + cos Atfδ + sin a sinBcZC.
This, and the preceding, also admit of a simple geometrical demonstration, by drawing the triangle and comparing the small increments in each case.
35. Again, since from any equation in spherical trigonometry another can be deduced by aid of the polar triangle, we get from the preceding dG= -cosδdA-cosαciB + sin A sinδdc∙
Corresponding formulae are obtained by an interchange of letters.
Again, the infinitesimals da, db, dA, dB ate connected by the equation da dB _ dA db tan a ^ t ^ taħ B tan A tan b
This follows immediately from the equation sin a sin B = sin A sin b.
36. These and the analogous formulae, when we adopt small differences instead of differentials, are of importance in astronomy in determining the errors in a computed distance arising from small errors in observation. They also enable us to determine the circumstances under which the most favourable observations are made, viz., those for which small errors in observation produce the least error in the required result.
The relations between the variations in the sides and angles of plane and spherical triangles were first treated of by Cotes, in his Estimatio Errorum in mixta Mathesi (1722).
(1) The values of — and — , when x, y, z are connected by two dx - dx equations of the form ∕(z,y,2)≈≈0>, <∕>(itf,y,ε) = 0, âre found to be dfdφ df dφ df dφ df dφ dy _ dx dz dz dx dz _ dy dx dx d y dx^~ dfdφ dfdφ^' dx~ df dφ df dφ ' da dy dy da dz dy dy dz
(2) If ∕(ω) = ≠(-y), where u and v are each functions of x and y, it is easily shown that du_dv dudυ θ dx dy dydx
(8) In a spherical triangle, if ⅞ x -- P be constant, and equal toκ, sin a the relations da + db do _ θ cos A cos B cos C - and cos A da + cos B db +cos C de =∕t 2 d (sin a sin b sin c)
can be readily established.
(4) More generally, it may be shown that, if κ also be supposed to vary, : + ~~~=o + —~p =tan A tan B tan Cd (—), cos A cos B cos C ∖ κ ) and cos Ada + cos B di + cosCdc=κd(κ sin a sin b sin c).
(5) If u be a function of ξ,η,f, and ∣=y+- , 1j =g+J f z x C=χ + —, show that y du du du . Mu , du .du
-√.⅛ + ,⅛ + .⅛Y
∖ dζ dξ d∙ηj Taylor's Theorem and Development of Functions.
37. We have already noticed that the development of functions by infinite series was a branch of analysis that rose into prominence during the latter portion of the 17th century.
The first series thus published were—that of Nicholas Mercator in his Logarithmotechnia (1668) for the expansion of log (Ι+ae), or what was then styled the area of an hyperbola (this he arrived at by the aid of Wallis’s method of quadratures); and that of James Gregory, in a letter to J. Collins, 1671, for the expansion of an arc in terms of its tangent. About the same time the first efforts of Newton’s genius were directed to this subject; and, aS we have already seen, he thus arrived at his binomial theorem, and other general expansions, such as those of sih x, cos x, e x , &c.
It was not, however, until many years after these discoveries that it was found that all such expansions may be regarded as particular cases of one general theorem. This theorem was discovered by Dr Brook Taylor, and published by him in 1715 in his Methodus Incrementorum.
38. Before proceeding to a consideration of this important series it should be observed that, in 16Θ4, John Bernoulli published, in the Acta Eruditorum, his well-known expansion under the title Additameniu/m effectionis omnium quadraturarum et rectificationum curvarum per seriem quandam generalissimam.
This series may be written as follows, slightly altering Bernoulli’s notation :—
∕ , ae 2 dy a? 3 d 2 y ,
Bernoulli obtained this result immediately by differentiation, by which process it can be easily verified.
This is the first general theorem on series that was discovered; and it -was easily shown by its author that the ordinary series, such as the expansions of log (1 +x), of sin x, and others, can be deduced from it.
This theorem of Bernoulli, however, is but a particular case oi Taylor’s, as will be shown subsequently.
39. Taylor arrived at his theorem as a particular case of another in finite differences,— a branch of the calculus treated of for the first time in his Meth. Increm. Introducing the modern notation, Taylor’s proof, with some modifications, is as follows.
Let f{x) be any function of x, and suppose x changed successively into x+∆x, ae + 2Aae, ir + 3∆τ, ... x+n∆χ∙, and let the functions
∕(x) > f( x + δ ≈), flx + 2∆ic), . . i ∕(x+ n∆x) be represented by V, Vi, Vz, · · y∙∙
Then we have y 1 -y=Ay, y. 2 -y 1 =Ly 1 , .. . y n -y*-ι≈∆y n -u ∆y 1 -∆y=∆ 2 y, ∆y 2 -∆y 1 =∆ 2 y i , ∆y n -∆y n -ι = ∆ 2 y n -ι, ∆ 2 y 1 - ∆ 2 y=∆ a y, &c.
The final result consists in expressing y n in terms of y, ∆y, ∆ 2 y, . . . ∆ n y . We have yn=yn-1 + ∆y n -1=y n -2 + 2∆y n -2 + Δ 2 y n -s.
In like manner, substituting y n -3 + Δyn-3 for y n -2, we get y n =y∏ - 3 + 3∆y n _ 3 + 3∆ 2 y n _ a + ∆ 3 y n - 3, the coefficients being the same as those in the expansion of (α + δ) 3 . Now, if we assume that the same law holds for any value n, it is readily seen by the method of mathematical induction, of which we have given an example in § 27, that it holds for the value immediately superior; and we thus get
τι.(π-l) . . m.Γn-l)(n-2) _ 1 , a .
y n =y+n∆y+ γ 2 ∆ 2 y + ’ -∆ 3 y+ ... + ∆"y.
40. This result can be readily established also by the principles of the symbolic calculus, a branch of the subject to which a short space will be devoted subsequently. We shall anticipate the consideration of that method by giving an application of it to the determination of the preceding result.
Regarding Δ as a symbol of operation, the equation y n =yn -1 + ∆y n -1 may be written y n =(1 + ∆)y n -1.
In like manner, y n -ι = (l + ∆)y n -2 .∙. y n =(1 + ∆)(1 + ∆)y n _ 2 = (1 + ∆) 2 y n -a, also y n =(1 + ∆) 3 y n - 3; and in general y n = (l + ∆) n y
≈^l + ⅞Δ+-∙fo~ 1 >Δ a + ... +Δ^ΐ/ ≈y+⅞∆y÷ n ∙^~ 1 ^∆ 2 y+ . . . ∆ n y.
1. Δ
41. If we suppose , h n∆x≈h. or n=—- , ∆x the equation becomes ., ,. 1 ,∆y h'.(h-∆x) ∆ a w .
∕(x + Λ)=y+⅛+— r g- i 5 +feix
If now, h being regarded as constant, we suppose n to increase, and consequently ∆x to diminish, indéfinîtély, we obtain, ön proceeding to the limit, /(a . + Ä ) = y + Ä g + '^ g + &c .
≈flf} +w+⅛rω+ i ⅛ r (*)+&*
This is called Taylor’s series.
42. In order to complete the investigation, it wi∏ be necessary to examine into the convergency or divergency of the series, and to obtain an expression for the remainder in it after any number of terms; this we shall immediately proceed to consider.
43. It may be observed that Taylor does not seem to have been aware of the great importance of his theorem, nor did he give any examples of its application. This probably accounts for the fact that so long a time elapsed before its real value Was discovered; and, although Stirling introduced a particular case of it in his Methodus Differentiatis (1717), it was not noticed in any of the English treatises on the calculus—such as Simpson’s Fluxions (1737), Emerson’s Fluxions (1743), Landen’s Residual Analysis (1764),—nor is it mentioned in the first edition of Mpntucla’s Hist, des Math., 1758. The theoréin is to be found in Euler’s Cal. Dif. (1755); but, although Euler makes extensive use of it, be made no reference to Taylor’s name in connexion with the series, and would appear to have given the theorem as his own, or rather perhaps to have connected it with Bernoulli’s series.
44. We may observe that Taylor also introduced into his Methodus Incrementorum, in thefluxional notation, a series which is the same as that of Bernoulli, already noticed. This led to a long and bitter controversy between them, in which Bernoulli’s son Nicholas and others also took part. In this Taylor was accused of plagiarism both with rèspéct to this theorem and to other theorems relative to the general theory of the centre of oscillation of bodies. It is remarkable that in this dispute no reference was made to Taylor’s own theorem, nor do the disputants seem to have been aware of its vast superiority to that around which the angry controversy was raised.
45. Taylor’s theorem seems never to have risen into dite prominence until its great value was pointed out by the illustrious Lagrangè, in the Berlin memoirs for 1772. Lagrange demonstrated the theorem by the principles of ordinary algebra. He made it the foundation of the method of series, and also of the differential calculus. He thus proposed to make the calculus a branch of ordinary algebra, and independent ôf all considerations of infinitely small quantities, and so to give it all the formal rigour of demonstration of the method of thé ancients.
46. Lagrange also was the first to placé Taylor’s théôrem on a satisfactory basis by finding an expression for the remainder of the series after any number of terms.
The following démonstration of this theorem of Lagrange depends on a single lemma, which may be thus stated. If a continuous function f (x) vanish when x = a, and also when x=b, then its derived function f'(x), if also continuons, iiιust also vanish for sonιe value of ± between a and b.
This is easily proved; for if∕'(*) dôés not Vanish for some Value of x between a and b, it must häVe always the éame Sign between these limits, and consequently fix) must constantly increase or constantly diminish as x passes by small increments from the value a to the value b ; but this is impossible, since f ix) vanishes for both limits. Now let R» represent the remainder after η terms in Taylor’S expaħ-sion, then writing X for x + y in that series, we have
∕(X)=∕U∙) + i2^ zω+ <2^y^) + ...
JL 1. Δ
+ ⅛⅛-7< n -b(≈) + R n , . · - (a), in which fix'), Tip) ff' ν >ix) are supposed finite and con tinuous for all values of the variable between X and x.
From the form of the terms included in R n it evidently may be written in the shape t ,-∣⅜⅛, where P is some function of X and x. Consequently we have ∕(X)- j∕(aj )√*^y'φ) + ... √p∕ cn - 1 >(≈) + (x-⅛ p i =0 . j . t ∖ . (3) .
jn_. )
Now, let z be substituted for* in every term in the preceding, with the exception of P, and let F(z) represent the reSuiting expression, we shall have
W=∕(X) - j ∕(2 ) + ⅞⅛) +. . . + j. . (γ), in which P has the same value as before.
Again, the right-hand side in this equation vanishes when s*≈X, .∙. F(X) = 0.
Also, from (3), the right-hand side vanishes when z±x;
.∙. F(*)≈0.
Accordingly, since the function F(z) vanishes when 2 ÷= X, and also whens=*, it follows from the preceding lemma that its derived fonction F'(⅛) also vanishes for some value Of z between the limits X ahd x.
Proceeding to obtain F ’ (s) by differentiation, it can be easily seen from equation (γ) that we have r <∙>- ¾i> ’ <≈>+⅞¾⅛∙
Consequently, for some value of z between x and X we must have ∕<">(2 ) = p .
Again, if 0 be a positive quantity less than unity, the expression *+0(X-*), by assigning a suitable value to 0, can be made cqτtal to any number intermediate between x and X.
Hence
P=∕< n >{* + 0(X-*)}, where 0 is some quantity >0 and <1.
Consequently, the remainder after n terms of Taylor’s series can be represented by
(X-*)» r
—∣^∕ w { a,+0 (χ - a5 )}∙
This is Lagrange’s form for the remainder. Substituting this value for K, i iiι (a), it becomes
∕(X)-∕W+ i ⅛⅛rM+¾irW+ ...
+ ¾Hr i ∕''^ υ w+-p7 i "∕ w {∙+*χ-*>!
Again, if h be substituted for X - x, the series becomes fix + h) =fix) + hf ’ (x) + &c. + /(»4 ι)(*) + £/(") ix + Oh ).
In this expression n may be any positive integer.
47. The last equation may be regarded as the most general form of Taylor’s theorem. We infer from it that the essential conditions for the application of Taylor’s theorem to the expansion of any function in a series are—that none of its derived functions should h n : become infinite, and that ∣-~ ∕<">(*+0λ) should become infinitely small when n becomes sufficiently large.
48. The remainder in Taylor’s senes admits, as was shown by Cauchy, of being written in the form ⅞5 ι ~y- ι ∕(">(i g+g⅞>.
I— 1
Another form was given by Dr Schlömilch, viz.
In some casés one or other of these latter values i⅛ preferable to Lagrange’s form.
49. Another remarkable mode' of determining the remainder in Taylor’s theorem was also giVefi by CaUchy. It is based on the following lemma, that if ιix) and∕(*) be two functions which remain continuous, as also their derived fonctions, between thé Values *ι and x 1 + h of x, and if also fix) does not become zero for any value of x between these limits, then
F(* 1 + ⅞) - F(* 1 ) _ F ’ (* 1 + Oh) . ∕ + *)' - ∕ (¾) ~fix i + βh) ’ where 0 is less than unity.
50. If in Taylor's series we make * + Λ=0, or h=> -it, We get
JL · Δ and henco
∕H=√(Q) +χfiχ) - -^- 2 f"iχ) + &c., a result which can be readily identified with Bernoulli’s series, given in § 38.
51. Again, if *==0, Taylor’s series becomes /(ä)=/(O) + V'(0) + ^/"(Ö) + &c.; or, as it may be written,
/(®)-/(0) + y∕'(0) +⅛∕"(0) + &C., in which fiθ), fiQ), ∕"(0), &c., represent the values of fix), f(x), fix), &c., when *=0.
This result is usually called Maclaurin’s series, having been giVeħ in his Fluxions (1742). It had, however, been previously published by Stirling in his Meth. Biff. (1717); but neither Stirling nor Maclaurin laid any claim to the theorem as being original, both referring it to Taylor.
. By substituting for∕(*) any of the elementary functions, such as sin x, cos x, log (l + *), we readily obtain their well-known expansions. It is to be noted that it is necessary in each case, for the validity of thé series i to show that the remainder after n terms becomes indefinitely small when n is taken sufficiently large.
52. The application of Taylor’s or of Maclaurin’s theorem becomes extremely troublesome in many cases, owing to the complexity of the successive derived fonctions. For example, if we seek to expand tan x by Maclaurin’s theorem, we have fix) = tan x, ∕'(aj) = sec⅛, fix) — 2 sec⅛ tan*, ∕'"(*) = 2 sec 4 *+4sec 2 * tan 2 *; and the subsequent derived functions iricrease in complexity. Similarly in the case of other elementary functions, such as sec x, cot *, &c.
53. The development of tail *, sec x, and many other functions is much facilitated by the aid of à system of numbers, introduced by James Bernoulli. These numbers are usually arrived at as follows. It is easily seen that the expansion of ∣ , in ascending powers of *, contains no odd power of * after the first, and that the two first terms of the expansion are 1 and - — . Accordingly
2
we may assume e*-l 2 1.2 i.2.3.4 + ∣± ”**
in which B 1 , B 2 , B 3 , &c., are constants. These constants are called Bernoulli’s numbers, and it can be shown without much difficulty that
Bι s =⅜ j ¾ ez ⅛V> B 3 ∙= t ⅛, B 4 = t ⅛, B s = ff ⅞, B 6 = -j7¾⅛, B 7 = ⅝, &c.
The complete investigation of the method of their determination is due to Euler. See his Calc. Diff., lib. ii. cap. 5.
In order to develop tan θ by aid of them, we write it in the form ten β ^τ(1 ^ r⅛) ’ whβre i ^ vz ι∙
Hence we find tan θ≈ 2 2 (2 2 - l)¾ θ + 2 4 (2 4 -l)-γ^ + 2 6 (2 6 - 1)⅞^ .... ∖ f- l± l∆
observing that
_1_ = _1 2
e a ≈+l e ac -l e 2x -l'
In like manner we get
. a 1 2 2 B 1 θ 2 4 B 2 θ 3 2 6 B 3 θ 5 .
∞ tβ -τ-<-⅛-⅛-- 1 -
a
Also, since cosec 0 = cot θ + tan —, we get cosβc 8 -1 + ⅛∑≡ + ¾^-υB,^ + 2(2*-l)iy + ... θ LA ∣A IA
For the completion of this investigation it would be necessary to consider the convergence or divergence of these series. This question would occupy too much space for treatment here.
54. The numbers B 1 , B 2 , ... were arrived at by James Bernoulli (Ars conjectandi, 1713, p. 97) in studying the summation of series of powers of the natural numbers 1, 2, 3. . . .
Thus, if Snr represent the sum of the series 1p+2p÷3p . . . +(n-l)P
Bernoulli proved that %p+i 7⅛p p P(P~ l)(,P-2)
8,l, ~Jψι~ 2^ + ⅜⅝ ,,,, ^ 1 ~ |£ ⅛∙ , -*÷ ∙∙∙
The numbers, B 1 , B 2 , B 3 . . . were defined by Bernoulli as being the coefficients of the first power of n in the expressions for Sn 2 , Sn 4 , Sn β , &c., respectively.
This series of Bernoulli may be established as follows
If each side of the identical equation , e nx _ 1
l + e≈ + e 2at + e 3 ≈ .... ⅛ e ("-ι)*=-- 1 - , e x - 1
be differentiated p times with respect to x, and we make z=0 in the result, we get 1p÷2p÷3p . . . +(n-l)r = Dp^^~^ , when∞≈0, where D stands for -A .
This may be written
SnP=Dp(-∑1) .
∖e x -l∕<o)
Again c nx - 1 c nχ - 1 x e x -1 X β x - 1
If φ(x) = ^J, and∕^= ae 1--A + A ae 2_&c., we get by Leibnitz’s theorem § 27,
SnP=∕(0)φ<p)(0) + jp∕(0)≠(p- 4 )(0) + ⅛∑2)∕'(0)≠(P-2)(0)+ . . .
Now it is easily seen that
Φ>*>(0)-⅞⅛:
and hence Bernoulli’s series follows immediately.
From the preceding we have wι{l∙-ι + 2 m ^ 1 + 3"∙^ 1 . . . +(2-l)*~ 1 }
m _ , m(m-l} = 2 m Z*' 1 -1 ’ R -m-2 _
2 + 1 .2 15j " · ’ ·
The function at the right hand side of this equation has been represented by φ (z, m), and called Bernoulli's function of the with order, by Professor Raabe (Grelle, xlii.).
Raabe has arrived at many remarkable properties of these functions, of which a few of the most elementary are here added.
φ(l -z, wι) = ( - l) w ≠(z, m).
≠(⅜, 2n)-(-l)»Ç^B B .
-^-φ(z, 2n) = 2nφ(z, 2n -1), where n > 1. dz ~. φ(z, 2n + l) = (2n + l)≠{z, 2n} + (-l)"-‰
, n ,. . ,, 2∣ 2n -1 i sin 2ιrz 1 sin 4ιrz sin 6τrz )
φ(z, 2n-l) = (-l)"J= r ∣^r + -42iΓτr + -θ^r+ · · φ
where z>0 and <1, and τι>l.
For their demonstration the reader is referred to Raabe’s memoir, as also to Schlomilch’s Compendium der Hähern Analysis.
It may be noted that the first fifteen of Bernoulli’s numbers were given by Euler in his Inst. Calc. Dif. P. 2, ch. 5. The next sixteen were calculated by Professor Rothe of Erlangen, and published by Ohm in Grelle, vol. xxii.; and thirty-one additional numbers have been recently calculated by Professor Adams, and published in the Proceedings of the British Association for 1877.
' The fractional part in each of these numbers was calculated by Professor Adams, by aid of Von Staudt’s theorem (Crelle, xxi.). This remarkable theorem is as follows. If 1, 2, a, a! . . . 2n, be all divisors of 2n, and if unity be added to each so as to form the series 2, 3, α + l, . . . 2τι + l, and of these the prime numbers 2, 3, p, p ’ . . . be selected, the fractional part of B n will be rυn i⅜ + T + 7 + F∙∙∙ l·
55. Several methods have been given for facilitating expansions by series, of which one of the most general and remarkable is that given by Arbogast in his Calcul des Dérivations (1800).
This is a method for expanding a function of , X x 2 1 x i . a+t T + c Γ2 + ⅛⅛3 +4c ∙ in a series of ascending powers of x.
Let tι=α+⅛A + c JA + d x +&c.
1 1. Δ 1. Δ∙ 0 and suppose φ(u) represents the required function.
Also, let
ψ(φ>)=A + Bγ+C ^ 2 + I) 1X3 + &c · ≡∕(0)+γ∕ ’ (0)+⅛∕'(0) + *c.∙, then we have A =∕ (0) =≈ φ(α).
Also, writing u', u' ’ , u"' i &c. instead of du d?u d 2 u dx ’ dx 2 ’ dx? ’ c ' ’ we obtain, by successive differentiation of the equation f(x} = Φ(u), f' (x)≈φ'(u}.u', f" (x') = φ'(u).u" +φ"(u).(u') 2 , f'" (x) = φ ’ (ιι).√" + 3φ"(u).u'.u"+ φ"'(u)(uf, f iτ (x) = φ'(u}. u* + ≠"( w)[4√ u"' + 3(√ ’) 2 ] + 6φ" ’ (u). (uγ. u' ’
+ ≠ lτ (tt).(√) 4 ,
Now, u, u', u", u ’ ", . . . obviously become a, b, c, d, . . . respectively, when x=0.
Accordingly
B = ∕'(0) = bφ'(a'), C =∕'(θ) = c≠'(α ) + δ 2 ≠"(α),
D =∕"(0) = dφ i (a) + 3bcφ"(a) + lPφ"'(a), &c.
From the mode of formation of these terms, they are seen to be each deduced from the preceding by an analogous law to that by which derived functions are deduced one from the other; and, as f'(x~), f"(x) . . . are deduced from f(x) by successive differentiation, so in like manner B, C, D, . . . are deduced from φ(u) by successive derivation ; where, after differentiation, a, b, c, &c., are substituted for du cPu s
* i > S’ -^∙ · · s °-
If this process of derivation be denoted by the letter δ, then
B = δ.A, C = δ.B, D = δ.C, &c.
From the preceding, we see that in forming the term δ . Φ(α) we take the derived function φ'{a), and multiply it by the next letter b, and similarly in other cases.
Thus δ. b =c, δ. c ≈d, . . .
δ. b m =mb m ^ 1 c, δ. c m =mc m ~ 1 d. . . .
Also δ. φ ’ (a)b = φ'(a)c + φ"(a)b 2 .
This gives the same value for C as that found before; D is derived from C in accordance with the same law; and so on. As an illustration of this method, we shall apply it to find a few terms in the expansion of sin (α+ J^ + cι ≤- + rf τ -^ + ⅛c.).
Here A = sin a, B = δ. sin a=b cos a,
C = δ.. b cos a = c cos a - b 2 sin a,
D = δ. G=d cos α-3δc sin α-δ 3 cos α,
E = δ . D=β cos α - (4δd + 3c 2 ) sin α-6δ 2 c cos α + δ 4 sin a.
Arbogast’s theorem has been treated somewhat differently by Professor De Morgan. Thus, suppose
≠(α 0 +α 1 x + α 2 z 2 ... + α n z n + ⅛c.) = A o + A 1 z+A 2 x 2 . .. + A n ae n + &c.; then, if we differentiate with respect to a n , we have φ ∖ a 0 +a 1 x . . .+⅛rmc>⅛⅛. . . + ⅞½∙ aj * + fc c .
CtCtn CCdjι dd∏
Hence we infer that, if m be less than n, we have = 0, (Ld·ti also . · . =⅞^- 0 =≠'(⅜), dre w dα w -ι dα 0 and cIcCq
The values of A 1 , A 2 . . . A n can be hence calculated (see De Morgan’s Differential and Integral Calculus, arts. 214-220).
56. Lagrange, in addition to having been the first to place Taylor’s series on a satisfactory basis, also enlarged the powers of analysis by a remarkable theorem which contains Taylor’s as a particular case. This, which is commonly called Lagrange’s Formula, first appeared in 1768 in the Berlin memoirs, and may be stated as follows :—
If z be connected with x and y by the equation z=x+yφ(z), then the expansion, in ascending powers of y, of any function F(z) may be thus written :—
F(z) = F(x)+f f (a5 )F(a ≈) +r ⅛ I [≠(z)l 2 F'(z) I
This result can be deduced from Maclaurin’s theorem, as was shown by Laplace, thus :—
Let u= F (ae), and we may write
-o 0 ÷Arn 0 ∙∙∙÷grn∙∙∙ where (^∖ , · · · represent the values of — . .
∖ ⅜7 0 ∖ d fJo ⅜ ⅜ 2 when we make y = 0 after differentiation.
We plainly have w 0 =F(x).
Also, it is easily seen by differentiation that
• <fe —Z— ——Z — dy dx ’ dy dx ’ writing Z for φ(z).
Again dy ∖ <tyj dy ∖ dx ) dx ∖ dy J dx ∖ dx J Hence we can deduce in like manner d?u ( d ∖ 2 ( du ∖
⅛s-(¾) (z ⅛p and in general fz»—
tty" ∖ dx) V dx)’
If now we suppose y=0, since Z reduces to φ(x), and — to F ’ (z), dx we get
(⅛) o →m≈>.(⅛V⅛⅛[≡p⅛)∣...
from which the series immediately follows.
For example, let z=x + - (z 2 -l); A then the expansion of z becomes 2 - 1+ ⅜(≈ s -i) + ι⅛(⅜) ’ ⅛(≈ ’- 1 > i ∙∙∙ + ⅛(⅜)" (⅛) ’ "⅛ s -υ"∙
Again, from our equation we get z≈-- ~ W+y 8 .
y y
Hence ⅛=(1 -2iry+z∕ 2 )-L
Consequently we have
(l-2xy+y≡)-i-l + ∣l A(≈≡-l). . .
÷⅛(W>->"∙∙
If we write this expansion in the form
(1 -2xy+y 2 Vl≈l + Xy + X 2 y 2 . . .+X n y n . . . wehava > χ,-∣⅛ (⅛)*(x>-l>∙.
The class of functions represented by X n was extensively studied by Legendre, to whose works the reader is referred for further development.
An expression for the remainder in Lagrange’s series in the form of a definite integral will be given further on.
57. Taylor’s series admits of ready extension to two or more variables; thus, if we change x into x + h in the equation u ≈ φ{x,y), we get, by Taylor’s theorem, , , , . , du h 2 d?u h 3 d?u 1 .
* ,+ *>Ι-t⅛+Γ2 ** + 1727S +
If now we change y into y+k, u or φ(x, y) becomes u + k^ + ½- ^⅛ + ⅛c.; dy 1.2 dy i , du, , du 1 ,. d 2 u 1 , k 2 d 3 u . . h - r becomes h- r - + hk , +⅞ τ-0 5-5-0+ &c · »
dx dx dxdy 1.2 dxdy i and accordingly we have - » , , * , du , du φ(x + h , y+k)≈u+h - 7 - + k -r- dx dy h 2 d 2 u , , d 2 u k 2 d 2 u t « - + =—- - r - 2 + hk -5-5- + v-5 5-5 + iιc ∙
1.2 dx 2 dxdy 1.2 dy i
By aid of Lagrange’s theorem in § 46 we can obtain an expression for the remainder of the series.
In like manner, if u≈φix,y,z), we get . , , ., . du , du ,du h 2 d 2 u
≠(≈ +4 "* + i ' s + z >-" + ⅛ + i ⅛ + ⅛ + ∏2
k 2 d 2 u l 2 d 2 u ,, d 2 u 1 7 , d 2 u 1 -, <Z 2 w 1 .
+ — s τ-5+r-5 ∙5-5 + hle ~rτ- + Μ -rτr + ι l l τr7- + lic ∙
1.2 dy 2 1.2 dz 2 dxdy dydz dzdx
The method can be readily extended to a function of any number of variables.
1. As an example of Maclaurin’s theorem, the first three terms d? 2oj 5 in the expansion of tan x are x + -5- + -τ^ ·
o 13
(2) Prove that tan-ι (a>+⅛)=tan-⅛+Λ sin z ≡--(Ä sin z) 2 ⅛⅛
1 Δ + (⅛si n2 )3⅛≡⅛- &c.
3 where z=cot^⅛.
θ) If x ’ y ma y I ,c eas fiy expanded in terms of x dx dx by the method of indeterminate coefficients.
(4) By similar methods the first four term in the expansion of
(1 + z) x in ascending powers of x are found to be fι-Λ + !2⅛-Z⅛b.
V 2 24 16 J
(5) Find the development of f 8in .^ ig - in ascending powers of sin x sin 2a√
x, the coefficients being expressed in Bernoulli’s numbers.
(6) Prove that Legendre’s function X n satisfies the differential equation d- a ≈ 2 ) ^-2x^ + w (n + l)X n = 0i also that
⅛±- 1 = (2n + l)X n + (2π-3)X n - 2 +(2n-7)X n -4 + ∙ · ∙. dx Indeterminate Forms.
58. Another important application of the infinitesimal method is to the determination of the true or limiting values of indeterminate expressions.
For example, if the fractionnel becomes of the form — , or —.
Φtx) 0 ∞
when x=a, the fraction is said to become indeterminate for that value of x.
In fact, the method of the evaluation of indetermina te forms may be regarded as the foundation of the differential calculus, since the determination of the derived function of any expression fix') reduces to finding the limiting value + when Λ = 0.
h We shall first consider the case where∕(α) = 0, and <∕>(α) «= 0. Here the true value of is that of ∙ 7 ^ flt ^ l ^? when 7t is evanescent.
Φ(α) φ(α + h)
But ∕( a + 7t ) = yW+W+<½) _ ∕>t g7t >
4>(α + h) φ(fli) + hψ'(a + θjh) “φ'(a + 0 1 Λ)
= Z1⅛ when h≈0. ΦW
Hence the limiting value of the fractio^ is in this case represented b y 4⅛.
Again, if be also of the form A, its true value is that of φ (a) 0
∕-⅛⅜; and so on.
≠"(α) ’
In general, if the order of the lowest derived functions which do not both vanish is n, then the true value of =4⅛- is that of .
φ(α) ≠<-)(α)
For example, the fraction ⅛ 0 f the form -θ
tan ay - t⅛n ax 0
whenx=j∕, to find its tnιe value.
Here fx)=x a sin ay - y a sip. ax, φ(x) = tan ay - tan ax, • ‘· f ∖ x ) = ax a ~ 1 sin ay - ay a cos ax , ≠'(α∙) = - a se&ax, accordingly the required value is represented by =y a - 1 (y cos ay - sin ay) cos 2 ay. Φ (y)
59. Again, to find the true value of=4⅛ if ∕(α) = ∞, ≠(α)-∞.
ΦW
1
Here > w bich i≡ 0 F the f° rra W » when x=a. βf)
Hence, by the former case, its limiting value is that of
≠ ’ (≈) i ä r
∕'(≈) ( ≠(≈) j
Suppose A to represent the limiting value in question and we have A-∙0⅛ A≡, or A = 4J4∙ f↑a) φ'(a)
Accordingly the true valpe of the indeterminate form is found in the same manner as that of the form — .
0
In the preceding, in dividing both sides of our equation by A, we have assumed that A is neither zero nor infinite. It can, however, be easily shown that the true value ip either of these cases is still . 1 . t f(a) that of j j; .
≠'(α)
60. Again, the expression fx) × φ(x) becomes indeterminate for any valüe of x which makes one of its factors zero and the other infinite. The expression, hpweγeγ, is readily reduced to the form ∙^-; for, if∕(α) = 0, and φ(a) = ∞, we have
∕(α) × ≠(α)=∕(α)÷-, which is of the form —.
Φ{a) · 0
Also, if the true value of be unity when φ[a) = ∞, then
≠(α)
∕W→(≈)-4 ⅞⅛ - i.p w -{ ⅞⅛ . ,p-L j .
This is of the form and its true value can in general be found as above. By this means the true value of/(a;) — φ(x) when /(ae) = ∞, and φ{f) = ∞ can be found.
61. The expression u v becomes indeterminate in some cases; ∙for suppose y=u v , then log y—v log u. This lafter product becomes indeterminate whenever one of its factors is zero and the other infinite.
(1) Let v=0, and log u=±∞; the latter equation requires cither u = ∞ , orκ=0. Consequently u ν becomes indeterminate for either of the forms 0° or ∞ 0 .
(2) Let ’ υ=±oo, log u=0 ; the latter equation giyes u=l, and the corresponding indeterminate forms are I® or 1^ 4 °.
62. In many cases the true value of an indeterminate form can be best determined by ordinary algebra or trigonometry. Thus, for example, the expression ∖ ∕a^+ax+x 2 - ∖ la^ - ax - α∙ β ∖ fa+x l - ∖ ∕a-x is of the form -θ when α=0. To find its true value we multiply by the complementary surds, when the expression becomes ∖ la j rx + ∖ ∕a-x ax + x t 'faP+ax+xP+'daP-ax-x^ x ’ the true value of which is plainly √α, when χ=O.
63. The differential calculus was applied for the first time to finding the true value of an indeterminate form by John Bernoulli, in the Aeta Eruditorum, 1704, when studying the problem of drawing the tangents at a multiple point on a curve. This problem, aS stated already in the Introduction, was started by Belle, as a crux for the advocates of the differential calculus. It may be here remarked that the determination of the tangents at a multiple point is generally much simpler by Cartesian coordinate geometry than by the method of the differential calculus.
A few elementary examples are added of the different classes of indeterminate forms here given.
, n ∣ e mx -e ma ∖ ,
- (1) u≈ -, r- , when x~=a.
I (≈-α) r l
Here f%) = f t"*- e ma , φ(x) ≈ (x - a) r , • ’ · ∕*(∞)= mcmx > Φ'{ χ ) =r ( x ~ α) r ^ 1 ·
Accordingly, when r>l , u=∞ ; when r=l , u=me ma ; when r<l,u=0
(2) .,⅛7⅞,⅜ w⅝ m ∣ .-..
(n 2 ^^) r
For r>l, u≈<⅛∙ i for r=1, «= -?; and for r< 1, u = 0.
2t
,< i ∖ aj p + 2cosaj-2
( 3) t≡ι⅛-'* h ∞*- 0 '
Since =1 when x=0, the true value of u, in this case, is tan x the same as that of ¾-⅜ff.ff⅛ffΓ ?· , and is easily seen to be ⅛ X*
(4) w=-‰ when ae=<».
e x
Here u= (~ ∖ ; but the true value of , when x = ∞, is easily ∖ e n ∕ e “ seen to be zero, consequently the true value of u is also zero.
(5) t4=^l + (1) w h en ®=τθ, and (2) when aj=*α>.
The true values are (1) u=l ; (2) ut≈e a .
(θ) ^fx 2 + ax - V≈ 2 + bx, when Xj= co,
This is of the form ∞ - ∞; its true value, however, is that of {a - b)x <t - ¾ a-b ∖ ∕x 1 + ax + ^fx 2 + bX /. a ∕. b 2
√ 1 + ^^ + √ 1 + V
(7 ) sin 2 3+ycos g ^→* when x=zy . τrue va j siu2ff> x n -y n
(8) φ ⅜(¾⅞w≡^0. True value, ⅛
sιn β aj
1
(9) x(l-a x ), when oα.
This is equivalent to ~ g when s = 0, and accordingly its true value is - log a. _
(10) Γ. , when ¾=0. True value,- r ⅛.
y x a , Maxima and Minima.
64. We have seen in th® Introduction that the question of finding the greatest and least values of an expression was, in the hands of Fermat, one of the first applications of the method of infinitesimals. We have also seen that the principle of his method had been previously stated correctly by Kepler, and is the same as that obtained by the differential calculus. We now proceed to a more general investigation on maxima and minima.
Let u represent the function, and x the variable, and suppose we have u =/(»).
Let a be a value of x corresponding to a τhqxiρvum or a minimum value of u, then for a maximam we must have ftp.) >fa + h), and fa) >fa - h), for small values of h ; and for a minimum, Λd)<faJ r k), a,n^fa)<fa-h).
Accordingly, in either case, fa + h) —fa) and fa - h) -/(«) must have the same siun. h being small.
But we have already seen that fa+h) -fa)≈hf(a) + ⅛∕''(α+ θh), fa - h) -fa) = - hf ’ {a) + f"(a - 0Ä), where θ is > 0 and < 1.
Now, when ∕ ’ (α) is finite, it is plain that fa+h)-fa) and fa - h) -fa) cannot have both the same sign, when h is very small, unless f(a)-0.
Accordingly, the roots of the equation f(x)≈0 furnish in general the values of x for which fx) has a maximum or a minimum value.
Also we have in this case—
⅛a fa + h) -fa) - j— - f' ∖ a f Qh),
*a f a -h)-f a )^-H-ffa-θh). 1 · ∆
Consequently, when /"(a) is negative, the corresponding value of fa) is a maximum; and when f"(a) is positive, fa) is a minimum.
If, however, f'(a) vanishes, along with ι ∕ y (α), it is readily seen that the corresponding value of u is neither a maximum nor a minimum unless /'"(a) also vanish.
In general, let f n ∖ a) be the first derived function that does not vanish; then, if n be odd, the corresponding value of u is neither a maximum nor a minimum: but if n be even, the corresponding value is a maximum when f n ∖ a) is negative and a minimum when it is positive.
These rules for distinguishing between maxima and minima were first given correctly by Maclaurin, in his Fluxions, ch. ix.
If the equation f{x) — 0 has no real solution, then flx) has no maximum or minimum value, and consequently is capable of having all possible values from + ∞ to - ∞.
We shall illustrate the preceding theory by applying it to a few simple cases.
(1) u≈aP+ax+b.
Here ^=2»+a= 0, , ∙. »=? - , ax ’ 2 ’
Again ^⅛=2. Since 4 ^his is a positive quantity, the function is a
, Λ τ , . . 1 · T. « 2 . 1 minimum when x= - -% . Its minimum value is b —— ; as is also ., . , ( a ∖ 2 , a 9 evident because u = I χ + — 1 + δ - — .
∕ ox ax i +2bφ + c
< 2 >
Here a» 2 + 2bx +c= u(a ’ xl i + 2b ’ x + cl).
Differentiate both sides, apd, since ^=0 for a maximum or a ttx minimum, we have ax + b≈ (α⅛ + b ’ )u.
Hence the roots of the quadratic (ab ’ - ba')x 2 + (αc' - ca')x + bel - cδ'=0 give the required solutions.
The corresponding values of u are given by the quadratic w 2 (δ' 2 - a ’ cl) + u(a<l + ca' - 2δδ') + b 3 - ac =0 .
If the roots of the quadratic in X bo imaginary, the proposed fraction has no maximum θΓ minimum value. When the roots are τeal, the fraction has one maximuni and one minimum value. These can be easily distinguished in any particular case. It is easily seen that to the greater root corresponds a minimum, qnd to the lesser a maximum value of the fraction, in general.
(3) w = tan » —⅛.
ττ dw , , dPu n „ ,
Here -y- = sec 2 » -1, - r -~ = 2 sec"» tan χ, ax ctx i C^"iL
- r ÷.≈2 sec 4 » - 4 sec 5 » tan 2 ».
dx i
Hence, for a maximum or a minimum we have sep ⅛=1,
Λ XI 1 O
. ’. tan » - 0; consequently + 0, and-^- 3 — 2, Accordingly the proposed has neither a maximum nor a minimum value.
... x < 4 ’ - γb3 ∙
The fraction - is a maximum or a minimum according as
1 + »® .
—is a minimum or a maximum, as is evident from the principle that, when u is a maximum, — is a minimum. But X + — is a U " *c maximum or a minimum when -L = 1, or »= ±1.
X“
Again, it is easily seen that the upper sign corresponds to a minimum and the lower to a maximum. We accordingly conclude that ⅛ is the maximum value of x , and - ⅜ its minimum value.
1+» 3 4
(5) The expression u = x x has its critical value when » = -L. e
65. Again, to find the maximum or minimum values of u, if w=∕(a), where z=φ(x).
Here g =f(≈)φ'M, and consequently the solutions of the problem are—(1) those given by φ'(x)≈Q, i.β., the maximum and minimum of z; (2) those given by∕(z)=o.
In many cases the values of z are restricted by the conditions of the problem to lie between given limits; accordingly in such cases no root of f ∖ z) ≈ 0 can furnish a real solution unless it lies between the given limiting values. This result will be illustrated in the following examples.
1. To find the maximum and minimum perpendicular from the focus on the tangent to an ellipse, the perpendicular p being expressed in terms of the radius vector r.
The expression for the perpendicular p, in terms of tlιe radius vector, is „ b 2 r dp ab 9 ∙'∙ ρ dr~(2a-r) 9 '
Accordinglyg=0 gives r=±α>; but these values are inadmissible, since r is restricted to lie between the values α(l +e) and α(l - e).
Consequently the only maximum and minimum values of p are those which correspond to the maximum and minimum values of r; i.e., α(l + e) and α(l-e).
2. To find in an ellipse the conjugate diameters whose sum is a maximum or a minimum.
If r and r' be two conjugate diameters, we have r 2 + r' 2 =α 2 + δ 2 , . ’ . u≈r + *√a a + δ 2 -τ' 2 .
The solutions accordingly are given,—(1) by the maximum and minimum values of r, and (2) by the equation
1- -. =0. f fa 9 + b 9 -r 2
The latter gives the equiconjugate diameters, the former the axes of the ellipse. It is easily seen that the former solution gives a maximum, the latter a minimum ; ns is also readily shown otherwise.
3. To find the position of a planet when brightest, its orbit and that of the earth being supposed circular, and to lie in the same plane.
Let S, E, P (fig. 4) be the positions of the centres of the sun, earth, and planet respectively. Let A0BD represent the section of the planet made by the plane SEP. Draw AB perpendicular to SP, and CD perpendicular to PE. Then ADB represents the illuminated half of the planet, and CBD the half visible from the earth. Accordingly the portion of the illuminated surface turned towards the earth is contained between two planes drawn respectively through AB
and CD perpendicular to SPE. This surface is projected into a crescent, the breadth of which is proportional to the versine of BPD, or to 1 +cos EPS.
Again, the brightness, depending on its distance from the earth and its position respecting the sun conjointly, will vary as
1 + cos EPS
PJE 2
Letα=ES, δ = PSJ, »—PE; then x i + b 9 -a 9 1 + cosEPS ae 2 + 2te + δ 2 -α 3 COsEPS ∙'∙ —W 2te’ --
Hence, neglecting a constant multiplier, we have
1 2δ α 2 -δ 2 U ~ x + x 2 ae 3 ‘
<Λ√ «G
Accordingly, the solutions of the problem correspond to—
(1) The maximum and minimum valuesof », i.e., a + b and a - b;
(2) The roots of the equation ^, = 0 , or of
» 2 + 4δ» - 3(α 2 - t a ) = 0; whence we get x = V3α 2 + b 2 - 2b; neglecting the negative root, which is inadmissible.
If b>a, *f3a 2 + b' i - 2b is negative, and accordingly this gives no solution in the case of an exterior planet.
For an interior planet we have a>b; and it remains to determine whether √3α 2 + b 2 - 2b lies between the maximum and minimum values of x, i.e., between a + b and a-b 1
Since a>b, it is immediately seen that χ∕3α 2 + ⅛ 2 -2δ is >a-b∙ The remaining condition requires a+b>^f3a 2 +b 2 -2b, or α + 3δ> V3α 2 + 5 2 , i.e., a 2 + 6ab + 9b 2 >3a 2 + b 2 , or ib 2 + 3ab>a 2 .
Hence we easily find b > ψ · We accordingly infer that this gives no real solution for a planet nearer to the sun than one-fourth of the earth’s distance. When this condition is fulfilled it is readily shown that the corresponding solution is a maximum, and that the solutions corresponding to x = a + b and x=a - b are both minimum solutions.
66. Many problems of maxima and minima contain two variables, which are connected by an equation of condition. Thus, to find the maximum or minimum values of φ(x, y), where x and y are connected by the relation
∕(m)-o.
Here we have
⅜ ⅜ dy =0 df + df ⅜=, 0 . dx + dy dx ’ dx dy dx
Accordingly the maximum and minimum solutions are obtained from the simultaneous equations and∕(∞, y)-0.
dx dy dy dx
More generally, if n +1 variables, x, x 1 , x 2 . . . x n , be connected by n equations
F 1 =0, F 2 =0, . . . F n ==0, and it be proposed to find the maximum or minimum value of a given function f(x, x 1 , x 2 . . . α⅛ 1 ) of these variablés, we have, by differentiation, the equations dx + dx i + . . . +^-dx n ≈Q, dx dx 1 dx n
<ZF 1 , <ZF 1 , , , <ZF 1 , n - τ -÷dx+- r - 1 dx 1 + . . . +- r -ldx n ≈0, dx dx 1 dx n ^ d χ + ^dx 1 + . . . + ^dx n ≈0, df dXγ dXn which give, on elimination, the determinant equation [table]
This joined with the given equations determines the system of values of x, x l , x 3 . . . x n for which the function may have a maximum or a minimum value.
Maxima and Minima for Functions of two or more Independent Variables,
67. Let u=φ{x, y}, then, as in § 64, if x and y are independent, the maximum or minimum value of u must satisfy the equations du λ j du n
— =0, and —=0.
dx dy
Suppose a⅛ and y 0 to be values of x and y which satisfy these equations; then, in order that they should correspond to a real maximum or minimum value of u, the expression
≠(⅞ + Λ∙, y 0 + k)-φ{x 0 , y 0 ) must have the same sign for all small values of h and k, as in the former case.
Again let A, B, C be the values of - --f j respectively, dx 2 dxdy dy 2 r . when α∙=x 0 and y = y 0 , Then, by § 57, φ(x 0 + h, y 0 + k)- φ(x 0 , y 0 ) = ⅜( AA 2 + 2BΛ⅛ + C⅛ 2 ) + &c.
But, when h and k are very small, the remainder of the expansion is, in general, very small in comparison with AΛ 2 +2B⅛⅛+C⅛ 2 ; and consequently the sign of φ(x 0 + h, y^- s rk)- φ{x 0 , yj) depends on that of Nh 2 + 2Bhk + Ck 2 , i, e., of (a ⅞ + W.+ ⅞ 2 ( a C.- b8 ) .
A
Now, in order that the latter should have the same sign for all small values of h and k, AC-B 2 must not be negative; i.e., ^r^7 · τ⅛ - (-τ~τ~ ) must not be negative. When this condition dx 2 dy 2 ∖ dxdy ∕ holds, the resulting value of u is a maximum .when A is negative, and a minimum when A is positive. The necessity for this condition was first established by Lagrange.
In the particular case where A = 0, B = 0, C = 0, then for a real maximum or minimum it is necessary that all the terms of the third degree in h and k in the expansion of φ{x 0 + h, y 0 + k) should also vanish, and that the quantity of the fourth degree should preserve the same sign for all values of h and k.
The preceding discussion admits of a simple geometrical interpretation by considering the surface represented by the equation z = φ{x, y) ; since it reduces to finding the points on the surface of maximum or minimum distance from the plane of xy.
68. Next let i4=ζ⅛(ic, y, z), whereas, y, zare independent variables. Here, as before, if x 0 , y 0 , z 0 correspond to a maximum or a minimum value of u they must satisfy the equations du θ —-0 — 0 dx ’ dy ’ dz
Accordingly we have φ{x 0 + h, y 0 + k, z 0 + Z)-≠(a⅛, y 0 , z 0 )
≈ i(AΛ 2 + B⅛ 2 + CZ 2 + 2F⅛Z + 2GAZ + 2HA⅛) + &c., where A, B, C, F, G, H represent the values of (Pu d 2 u d 2 u d 2 u d 2 u d 2 u dx 2 ’ dy 2 ’ dz 2 ’ dydz * dzdx ’ dxdy respectively, whenx=ir 0 , y=y 0 , z=z 0 .
Now, as in the former case, in order that u should have a maximum or a minimum value, it is necessary that Ah 2 + Bfc 2 + CZ s + 2F⅛Z + 2GZÄ + 2HÄ*
should preserve the same sign for all small values of h, k, and Z. If we multiply by A, this expression may be written
(AÄ+H⅛ + G l) 2 + (AB - H 2 )F + 2(AF - GH)⅛Z + (AC - G 2 )Z 2 .
Consequently the sum of the last three terms must be always positive.
Hence, in order that the expression in question should be positive for all small values of Λ, k, and l, we must have [table]
This result is also due to Lagrange.
The corresponding conditions for the case of four or more independent variables can be likewise determined, and are readily expressed in the form of a series of determinants. See (Quarterly Journal of Mathematics, 1872, p. 48.
(1) To find the maximum or minimum value of the function ax 2 + 2hxy + by 2 + 2goe + 2fy + c.
It is easily seen that when h 2 >ab, there is neither a maximum nor a minimum value. When ab>h 2 , and a>c, we obtain, as the minimum value, [table]
ι 1 2) Similarly it can be shown that the maxima or minima values of ax 2 + by 2 + 2hxy + 2gx+2fy+c a'x 2 + b'y 2 + 2h'xy + 2fx + 2f'y +d > are the roots of the cubic equation [table]
(3) Of all triangular pyramids standing on a given triangular base, and of given altitude, find that whose surface is the least.
Tangents and Normals to Curves.
69. The infinitesimal calculus furnishes, as we have seen, a general method of finding the tangentat any point in a curve whose equation is given. For example, let y=f(x) be the equation, in Cartesian coordinates, to any curve; and suppose {x, y), (x 1 , y l ) to be the coordinates of two points P,Q in the curve (fig. 5), and (X, Y) those of any point on the line passing through these points, then the equation of the line is
Y-y=(X-α)^.
If now the point Q be supposed to approach P, and ultimately to coincide with it, the line becomes the tangent PT at the point P, and its equation becomes
Y- 2 ∕=(X-x) J.
For example, in the curve represented by x m ≈ay n , , dw my we have -√ δ =- > dx nx and the equation of the tangent at the point x, y is m n —=m -n . x y
This furnishes a simple construction for the tangent at any point on a parabolic curve. (Compare Ricci’s construction given in the Introduction, p. 7.)
If the equation of the curve be given in the form w=√U,y) = 0, , du du dy „ we have -=-+- r - -7 = 0, dx dy dx and the equation of the tangent is
< χ -*> a+< γ -≈'> ⅛- 0 ∙
70. Again, the normal at the point {x , y), being perpendicular to the tangent, has for its equation du du ... .
⅛(x - x )-<fe(γ -≈ ’ >∙
the curve being referred to rectangular axes of coordinates.
71. The line TM in fig. 6 is usually called the subtangent and RM the subnormal. It is easily seen that
MR-≡⅛,TM-X-i,⅛
dx dy a dy dx as it may otherwise be written.
Again, if the angle PTM = ≠, we have tan ≠=⅛; and dx the length of the normal
PR=ysβc≠ = ^l+^ i j also that of the tangent
PT = r√ cosec φ-y(i+⅛Jγ.
72. In general, if the equation of a curve be given in terms of any two variable coordinates, the position of the tangent at any point can be determined by finding the ultimate ratio of the corresponding elementary variations of the coordinates at the point. Newton gave, in his Opuscula, several applications of such systems of coordinates. In particular, it may be noticed that he considered the case of what are now called bifocal curves, i.e., where the equation is expressed in terms of the distances from two fixed points. Newton illustrated his method by finding the tangent to a Cartesian oval, styled by him an ellipse of the second order. The same problem, in a more general case, was studied by Leibnitz {Ac. Brud., 1693), who gave a method of drawing tangents to curves given in terms of the distances from any number of fixed points.
73. At a double point on a curve (see Cukve, vol. vi. p. 719), we have — = 0 and ⅛=0: and at such a point becomes inde-
dx dy dx dιι terminate, —being of the form — , since ⅛= - ⅛f · o 0 dx du dy du dx
Applying the method of § 58, the true value of becomes in this case that of Ty d' 2 u d i u dy dx i dxdy dx d' i u d i u dy ’ dxdy dy i dx d 2 u + d 2 u dy i e dy _ dx 2 dxdy dx ’ dx d i u d' 2 u dy' dxd<j dy' 2 dx
Hence ⅛ 2 f⅛'V= 0 dx 2 dxdy dxdy i ∖ dx)
The roots of this equation in ⅛ give the tangents to the two branches of the curve at the double point.
Double points are distinguished into three classes, according as the roots of this equation are (1) real and unequal, (2) real and equal, or (3) imaginary, ‰e., as(^-Yis >, =, or <0.
∖ dxdy) dx 2 dy 2
Of these the first are called nodes, the second cusps, and the third conjugate points. They are frequently also styled by Professor Cayley’s nomenclature as crunodes, spinodes, and acnodes. See vol. vi. p. 723.
74. In the general discussion of curves it is usually more convenient to refer them to a system of trilinear coordinates (see vol. vi. p. 719), in which the position of a point is determined by its distances from three fixed lines. The equations of curves in system are homogeneous.
Again, if {x, y, z), (a/, y', z') denote two points in such a system, the coordinates of any point on the line joining these points may be represented by λx + κx', Ky + κf, λz+κz'.
Hence, to determine the points in which the line joining x, y, z to as', f, ⅛ intersects a curve of the nth degree, we substitute λx + κa∕, λy + κy , , λz + κz , for x, y, z in the equation of the curve, « = 0; then by Taylor’s theorem (§ 57) the result may be written _ „ , ( ,du .du . du ∖ λ n u + λ n ^ 1 κ ( x ’ - r +y ’ - r +z , - γ ) ∖ dx 9 dy dz J or λ n α + λ"^ 1 κ∆w + -=— 5 - ∆ 2 u + ⅛c., where Δ stands for the symbol of operation (xl^-+y' 4-) · ∖ dx 9 dy dz)
The roots of this equation in — determine the coordinates of the κ
points of intersection of the line and the curve.
If the point x, y, z lie on the curve, we have u=0 ; if in addition we have , du ,du , du λ x - r + y - r +√ — =0 , or Δw = 0, dx. dy dz then a second point of intersection of the line with the curve will be consecutive to x,y,z ∖ and ∆w = 0 is the equation to the tangent at the point x, y, z.
Again, if the latter expression ∆u vanish identically, the point aj, y, z is a double point on the curve; or, in other words, every line passing through it meets two branches of the curve there. The equation Δ¼=0 is in this case that of the pair of tangent lines at this point to these two branches.
This method is evidently susceptible of much extension.
Asymptotes.
75. The method of the calculus furnishes a ready mode of determining the asymptotes to algebraic curves. By an asymptote we understand a tangent whose point of contact is situated at an infinite distance.
To find the asymptotes to a curve of the nth degree, we suppose its equation written in the form n n ÷Wn-l + ‰-2÷ · · · + u∙ι + uι + u<] ~ 0, where u n is a homogeneous expression of the nth degree in x and y, &e.
Again, writing ‰=z n ∕ 0 θθ , u n -ι = x n ~f 1 θθ , &c., the equation becomes
≈≈ n ∕o (⅜)+*-¼ (⅜)+ a ="^⅜ (i) +&c · =0 ·
Let y≈κx + v be the equation of any right line; then, to find its points of intersection with the curve, we substitute κ + -L for 7- in the preceding equation, and, after expansion by Taylor’s theorem, we arrange according to powers of x ; this gives
α n ∕o(κ ) + a ≈"^ 1 {Λ< ,f ) + y 7o(κ)}
+ {Λ(κ) + √ ’ ι(κ) + + &c · “0 ∙
Now if the line y≈κx + v be an asymptote, two of the roots of this equation in x must be infinite, and consequently we have
∕ u (κ) = 0 > and ∕ 1 (κ) + √' 0 (*) = θ ·
If κ' be a root of ∕ 0 (κ) = 0, the corresponding value of v is i⅛⅛', Λ(*') , and the equation y=κ'x- ⅛⅛χ
∕√κ)
represents an asymptote.
If∕ 1 (√) = 0, i.ei, if u n - ∖ and w n have a common factor y-κ'x, the line y≈ κ'x is an asymptote.
To each root of ∕ 0 (κ) = O corresponds an asymptote, and accordingly every curve of the nth degree has in general n asymptotes, real or imaginary. If the equation of the curye contains no tenus of the degree n- 1, the n asymptotes are represented by the equation z⅛=0.
In the case when ∕ 0 (κ) has a pair of roots each equal to √, then y ’ (κ ') = 0, and the corresponding value of vis, in general, infinite. In such cases the corresponding asymptote is situated at infinity. The parabola is the simplest case of this, having the line at infinity for its asymptote. Branches of this class belonging to a curve are called parabolic, while a branch having an asymptote within a measurable distance is called hyperbolic.
It is easy to establish an analogous method for finding asymptotes to curves whose equations are given in polar coordinates,
The equations to the real asymptotes in the following curves are easily found by the above method.
(1) z 2 y 2 = α 2 (.≈ 2 + y s ) + i> 4 . Ans. ae=±oe, y=±a.
(2) x 2 y 2 = a 2 (x 2 -y 2 ) + b ∖ x+y). ,, y + a=0 ∣ y-a≈Q.
(3) x 2 -x 2 y-{a + c)x 2 + axy + 2a 2 y + d=Q. ,, x + g,≈0,x- 2α = 0
x-y-c≈d.
(4) Prove that the asymptotes to a curve of the third degree meet the curve in points which lie on a right line.
(5) Show that the curve ae 3 - uxy + aby = 0 has a parabolic asymptote, and find its equation.
Curvature, Evolutes, Pointe of Inflexion.
76. The word curvature indicates deviation from a right line, the curvature at any point on a curve being greater or less according as it deviates more or less rapidly frqm the tangent at the point.
The curvature at any point on a curve is obtained by determining the circle which has the same curvature as that of the curve at the point. Let ds be an indefinitely small element of the curve, and dφ the angle between the tangents at its extremities, then ⅛ represents the radius of the circle which has the same curvature.
ÉL is accordingly called the radius of curvature of the curve at the dφ point. The circle is called the circle of curvature, and its centre the centre of curvature, corresponding to the point on the curve. Denoting the radius of the circle of curvatqrç by p, we have ds P dφ '
Again, if χ, y be the coordinates of the point, and φ be measured from the axis of x, then, since ds is the limit of the hypothenusp of a right-angled triangle of which dx, dy are the limits of the sides, we have tan≠-⅛f . · ⅛ sec 2 <⅛ ⅛≈ see 2 0 . ⅛.≡≤≠
dx ’ ‘ ’ dx 2 $ d x $ ds dx p
I 1 √⅛VH Hence p -⅛≠= i U2±.
djy <Py da? dχi
This expression for the radius of curvature was given by John Bernoulli {Acta Eruditorum, 1701).
The radius of curvature becomes infinite at a point for which
Such points are styled points of inflexion on the curve, and the tangent at a point pf inflexion is called a stationary tangent (vol. vi. p. 719). Other expressions for the radius of curvature can be readily obtained.
For instance, since j dx , . du cos , and sin φ = ~ , as r ds if the arc be taken as the independent variable, we have » . dφ d 2 x dφ d?y
- sin φ , çoa φ -~≈~ , ds ds i ds ds 2
. 1 - ⅜ ∕∕rf¾ ∖ 2 Hd 2 y ∖ 2
· 7~rfΓ= √ + ⅛J ·
Again, if p be the length °f thp perpendicular drawn from the origin on the tangent at a point whose distance from the origin is r, the radius of eurvatuya at the point j? given by the equation dr p≈r- 3 - . . dp
This value of ρ can be readily established from geometrical considerations, and is frequently useful, more especially in applications of the calculus to physical astronomy.
77. If the centre of curvature for each point on a plane curve be taken, we get a new curve called its evolute. Also, with respect to the evolute, the original curve is called an involute, and may be described from its evolute by the unrolling a stretched string supposed wound round the evolute. In this motion each point on the string describes an involute to the curve. The curves of the system thus described are said to be parallel. Again, from its definition, it is plain that the evolute of a curve is the locus of the points of intersection of the normals drawn at consecutive points on the curve.
78. Contact of Curves.— Suppose two curves, represented by the equations y=f{x) and y≈φ{x), to haye a point {x, y y ) in common, then f{x') = φ{x').
Let x + h be substituted for χ in both equations, and suppose y i and y, 2 to be the corresponding ordinates, then h 2 y l =√(*+&) =∕(≈)+ W +j⅛∕"(*)+&c., h 2 y 2 = Φ{ χ +A)= φ{ χ }+hφ'{ χ }+ ι - 5 φ"{x ∖ +&c.
JL. Λ
.∙∙ yι-y 2 =Ajr(^≠ ’ (⅛∣ f⅛j∕ ’ (≈)-≠"(≈)j+kβ∙
Now, if f'{x} = Φ'{x), we have ½ 2 ( ) hi i )
yι-y i = 17 2 ∖ f"^~Φ"^ ∣ + T‰ j ∕ ’ "(*) - ≠"'(*) j·+&c., and the curves have a common tangent. In this case the curves have a contact of the first order, and when h is small the difference between the ordinates y 1 and y 2 is a small quantity of the second order.
If in addition f'{x) *= φ' ’ {x), we have yi - y 2 = 1 - x - 3 j f ’ "(χ) - Φ"Xχ) j + &c.
In this case the difference of the ordinates is a small quantity of the third order; and the curves are said to have a contact of the second order, and approach indefinitely nearer to each other at the point of contact than in the former case.
Also, since y 1 - y 2 changes its sign with that of h, the curves intersect, as well as touch, at the point qf contact.
If, moreover, f"'{x} = Φ"'{ χ ), the curves have a contact of the third order.
In general, if f(x) = φ{x), f ’ (x) = φ ’ {φ) f f"{χ)≈φ"{x) . . . . ∕(n )(x) = <∕>(n )(x), the curves are said to have a contact of the uth order at the point.
It is plain from what precedes that, if two çurves have a contact of the nth order, no curve having with either a contact of a lower order can pass between them.
We shall illustrate this theory of the contact of curves by finding the circle which has a contact of the second order with the curve y=∕(≈) at the point {x, y).
Suppose («- a)^+(y-J3) 2 =β≡ i to be the equation of the circle, then, by the preceding, ⅛ qnd ⅛¾ must be the same for the circle dx dx 2 and for the curve at the point.
Differentiating twice successively the equation of the circle we get dy χ-u+{y-β) ⅛≈ 0 < 1+ (⅛y + (,. i )⅛= 0 .
b÷(l)T Jta>∞ + WL.
(sj
This agrees with the value for the radius of curvature found in § 76, and shows, as is indeed evident, that the circle of curvature is the circle having a contact of the second order at the point ip which it touches the curve.
Again, if x, y be eliminated between the preceding differential equations and that of the curve, the resulting equation in a, β is that of the evolute of the curve.
From what has been shown above, if the equation of a curve contain n arbitrary coefficients, we can in general determine their values so that the curve shall have a contact of the order n - 1 with a given curve at any point; for the coefficients can be determined so that y, . - —⅛ shall have the same values for the dx dx 2 ’ dafi~ ∖ two curves at the point.
The curve thus determined haying a contact of the highest order with a given curve at any point is called an osculating curve. For instance, as the equation of a right line contains but two independent constants, it admits in general of a contact of the first degree only. Again, the equation of a circle has three independent constants, and accordingly the circle admits in general of a contact of the second degree with a curve at any point. The parabola has four independent constants, and consequently admits of a contact of the third ord v er; and so on.
Again, introducing the additional condition ∕(")(αr) = ≠(")(aj), a finite number of points is seen to exist at which the osculating curve has a contact one degree higher; thus a tangent may have contact of the second order, an osculating circle contact of the third order, and so on.
In the case of à right line, we have
≠H=∕(*), φ'(x')≈f(x), φ"(x)=f' t (x), where φ(x)=ax+b, .'. φ"(x} = 0. This agrees with the condition found for a point of inflexion in § 76. The problem of contact admits of being considered also from a geometrical point of view, i.e., from the consideration of the number of consecutive points of intersection of two curves.
79. The discussion of evolutes and involutes originated with Huygens, in his celebrated work, Horologium Oscillatorium (1673), published before the invention of the calculus. Huygens’s investigation is purely geometrical. The definition of the osculating circle was first given by Leibnitz, in the Acta Eruditorum, 1686, where he pointed out its great importance in the study of curves. Newton, in his Principia, makes frequent use of the theory of the radius of curvature, and of its connexion with evolutes.
Newton also observed that the radius of curvature becomes infinite at a point of inflexion, and vanishes at a cusp—called by him pιιnctum rectitudinis, and punctum curvaturae ivflnitoe, respectively. See Opμsc,, i. pp. 121, 122, ed∙ Cast.
It is worthy of remark that Sluze, in his Mcsolabum seu duse mediae proportionates, kc. (1659), pointed out a general method for the determination of points of inflexion (puncta flexus contrarii), by reducing it to a question of maxima and minima, viz., to finding when the intercept made by the tangent, measured along any axis from a fixed point on it, is a maximum or a minimum. This methqd hθ applied successfully to the conchoid of Nicomedes.
(1) It is easily found as above that the radius of curvature at (ä^ I i^^)^∙ 5 any point on the curve 3α 2 y=jc 4 is equal to p≈> o · 4 ! , * ΔCL X
(2) The following expressiori for the radius of curvature in polar coordinates,—
HOT
can be easily deduced.
(3) If 2f = ∙⅛ f this becomes l>+(OT ρ ~ ePu , u - + dfi
(4) Hence at a point of inflexion we have
.
u+ d^- a -
(5) The origin is a point of inflexion on the curve represented by the equatjon u 3 + u,=0.
(6) The length of the radius of curvature at the origin in the curve r=a sin nt) is ∖ na.
(7) If on the tangent to a curve a constant length be measured from the point of contact, the normal to the locus of the points thus taken passes through the corresponding centre of curvature of the proposed curve.
(8) In the ellipse ^L⅛∣L,= 1, if we take ae=a cos φ, y=b sin φ, q j δ 0 δ the coordinates a, β of the centre of curvature of any point are given by the equations α .≤z≤∞ s ∙≠ 1 e .⅛i⅛ a p
(9) At a cusp (compare § 73) the radius of curvature is zero for both branches.
(10) In some cases two branches of the same curve may have a contact of the second or of a higher order. For instance, it. is easy to show that at the origin two branches of the curvç
y 2 -0a¾+⅛*-dc 5 ⅛0
have equal finite radii of curvature.
Envelopes.
80. If we suppose a series of different values given to a in the equation
∕(x, y, α)==0, then for each value we get a distinct curve, and the above equation may be regarded as representing an indefinite number of curves, a single determinate curve corresponding to each distinct value of a, provided a enters into the equation in a rational form only.
If now we regard the parameter a as varying continuously, and consider the two curves /(«, y, α) = 0, f(x, y, a + Δa) = 0, then the coordinates of their points of intersection satisfy each of these equations, and therefore also satisfy the equation f(x, y, a + ∆g) -∕(⅛∙, y, α) θ
∆α
Now, in the limit, when ∆α is infinitely small, the latter equation becomes df<f, y , α) q da
Hence the locus of the points of ultimate intersection for the entire system of curves represented by∕(x, y, α)≈0 is obtained by eliminating a between the equations
∕(a⅞ y, α) = 0 and ⅛ ig, V∙ g )≈Q.
dα
This locus is called the envelope of the system, and it can be easily seen that it is touched by every curve of the system.
For instance, suppose L, M, N to be given functions of x and y, and a a parameter, to find the envelope of the system of curves represented by the equation
Lα 2 + 2Mα + N = 0.
Here f(x, y, α)-∙ Lβ a + 2Mα + N;
.∙. ⅛⅛J>.2U+2M.
da
Consequently the envelope is the curve represented by the equation LN = M?.
For example, if L, M, N represent right lines, the envelope of the moving line
Lα 2 + 2Mα + N = 0, is the conic LN - AI 2 .
In general, if the equation of the moving curve be of the form P 0 α n + P 1 a"^ 1 + P 2 a"- 2 + . . . + P n =≈0, where P o , P j , P 2 . . . P ra are given functions of x and y, the envelope is obtained by the elimination of a between the proposed equation and its derived equation nP 0 ft n -1 + (n - l)P 1 α n - 2 + &c. = 0.
It is accordingly represented by the condition that the equation in a should have equal roots; this condition is called the discriminant of the equation. For examples see Salmon’s Higher Plane Curves, Arts. 85, 86.
81. In many cases the equation of the moving curve is of the form f(x, y, a, jβ) = O, where the parameters a, β are connected by an equatiop φ(a, β) = 0.
In this case we regard β as a function of «, and thus wθ get by differentiation df_ + ÿ_ ^L 0 ⅛ + ⅛ ^-=o∙ da aβ da ’ dμdβ da " ’ consequently, if we make
⅛. i ⅛,we g et⅛- λ ⅛, da da dβ dβ and the required envelope is obtained by the elimination of a, β, λ between these and the two given equations.
For example, let it be proposed to find the envelope of a line of given length (a), whose extremities move along two fixed rectangular axes.
Here, taking the fixed lines for coordinate axes, and denoting the intercepts by a and β, we have
Λ + ‰l, and α 2 + ∕3 2 = αθ . a β
Hence -⅞ = λa, -⅛==λβj a. p from which we get
λ = ⅜, √∙ f* 3 T⅛ ∕3 2 ^a 2 y, a?
and the equation of the envelope is
α⅛ + yδ = αi.
This envelope was discμssβd by John Bernoulli in the Acta Erud., 1692.
Again, to find the equation to the evolute of an ellipse, regarded as the envelope of its normals. Here we have the equations a 2Ä_ 6 2Ä ≈ α 2. δ 2 and ±* + g = ι a β ’ α- b 2 where a, β are the coordinates of a point on the ellipse. Hence g¾ i α ⅛∕
α 2 * α 2 ’ 0 2 6 2 ’
and we easily obtain ⅝s the required equation (axft + (δyj^= (a 2 - δ' 2 )i The preceding method can be readily extended to the general case in which the equation of the moving curve contains any number n of variable parameters, which are connected by n — 1 equations of condition.
82. The theory of envelopes, or of ultimate intersections, may be said to have originated with the investigations of Huygens on evolutes, already referred to, and those of Tschirnhausen on caustics (Acta Eruditorum, 1682). These authors, however, merely treated geometrically a few cases of moving right lines, and did not give any general method for the investigation of such problems. Leibnitz was the first who gave a general process for the solution of this class of questions (Acta Eruditorum, 1692, 1694). His method does not differ in any material respect from that here given.
(1) To find the envelope of the parabolas described by a projectile discharged from a given point with a given velocity, but at different angles of elevation.
If e be the angle of elevation, and h the height due to the initial velocity, the equation of the parabolic path is y i x=∙y tan e —~ — 5 - .
4/i cos∙ i e
Let tan c => a, and the equation becomes z=α 2 ∕-g(l + α 2 ), or^ + 3-α 2 ∕+α 2 g = 0.
Consequently the equation to the required envelope is y 2 =4Jι(hx), which represents a parabola.
This problem is the first that was brought forward on the locus of the ultimate intersection of curved lines. It was proposed by Duillier to John Bernoulli, who solved it, but not by any general method (Gommer. Epist. Leib, et Bern., vol. i. p. 17).
(2) To find the envelope of the system of conics represented by the equation aj 2 y 2 ,
—+ — 1 , a a — n where a is a variable parameter. Proceeding as before we get as the equation to the envelope (x±*fn) 2 + y 2 =0. Hence we infer that a system of confocal conics may be regarded as inscribed in the same imaginary quadrilateral.
(3) Find the envelope of the plane X y z _
- 7 - + -+ 1
t m n in which the parameters I, m, n are connected by the equation lmn=a 3 . Ans. 27 xyz=a 3 .
(4) A right line revolves with a uniform angular velocity, while one of its points moves uniformly along a fixed right line, prove that its envelope is a cycloid.
Symbolic Methods.
83. The analogy between successive differentiation and ordinary exponentials was perceived by Leibnitz and the early writers on the calculus, and afterwards more especially by Lagrange (Mém. Acad. Berlin, 1772). Arbogast was, however, the first to separate the symbol of operation from that of quantity in a differential equation (Calcul des Dérivations, 1800). The first writers who appear to have given correct rales on the subject of operations were François, Ann. des Math., 1812, and Servois, in the same journal, in 1814. Servois more especially exhibited the principles on which the legitimacy of the separation of the symbols of operation from those of quantity depends; and, making a separate calculus of functions out of those properties, he succeeded in proving that differences, differentiations, and multiplications by any factors which are independent of the variable, may be employed as if the symbols of operation were ordinary algebraic quantities. Hence has arisen a new method of considering the principles and processes of the calculus, called the symbolic method, or the calculus of operations.
In this method is written in the form and the symbol is regarded in the light of an operation, supposed to be made on the function u according to the established principles of differentiation·
Again ⅛ ( β+ ,).(⅛). + (ff ( 1 ), ai ≡°∙ (⅛)*(⅛)' t ∙√⅛n · · · < 2 >∙
And, if u be a function of x and y,
(⅛)(⅜)-(i)(±)— »
Hence we observe that the symbolsand ~ operate and are combined according to the same laws as ordinary algebraic symbols of quantity, such as a and b ; and we can readily infer that the theorems in ordinary algebra (compare Algebra, vol. i. p. 519, § 8, 9) which depend solely on such laws of combination are capable of being extended to similar theorems depending on the symbols — and —, dx dy or on the symbol and any constant a. Such results are in general capable of extension to any symbols that are subject to the same laws of combination.
The law embodied in equation (1) is called the distributive law; the second, in (2), is called the index or exponential law; and the third, in (3), the commutative law.
It is conveuient to denote the preceding symbols by single letters. Accordingly we may suppose the symbol to be represented by D, (LX and — by D ’, &c.
dy
In general, if π, p denote two symbols of operation such that τr(u + v) = ∙πu + τrv , p(u + v) = pU + pv, πpu = pπu , ιr m π n u = π' n ^~ n u , then the symbols ιr, p possess the distributive, commutative, and exponential properties.
For example, suppose Ea represent the operation of changing x into x + h in any function of x, i,e,, suppose Ea≠(≈) = φ(x + h). Then Ea I φ(x) + ψ(x) I = φ(x + Ä) + φ(x + Ä) = Ea≠(x) + E λ ψ(a!). Moreover, E* denoting the operation of changing x into x + k, we have ‰φ(x'}≈φ(x+k), ∙ ’. EA.E*≠(aj) = EA≠(a3 + ⅛) = ≠(a5+Λ + ⅛).
In like manner EaEa≠(≈) = φ(x + h + ⅛) = EA+*≠(aj), . , . EaE*≠(o!) = EaEa≠(≈). Hence the symbols Ea, E* are commutative.
Also the equation
E⅛EA≠(iP) = Ea+*≠(S!) may be written, symbolically, thus :— E⅛Ea=Ea+* .
This shows that the symbol Ea is of the nature of an exponential; and may be written in the form E i .
84. This symbol can also be connected with Taylor’s expansion. Thus, if we separate the symbols of operation from those of quantity in Taylor’s theorem, it may be written
∕(y , + A)=(l + λD + ⅛D 2 + r -‰D 2 + . . . W) = e* D /(ae)·
∖ 1. Δ l.∆.o J
Accordingly, although we can give no direct meaning to the symbol e AD , except as the representative of the symbolic expansion
Λ 2 h n
1 + W + ⅛D 2 ÷ ... +⅛D-...
we may from the preceding section regard it as equivalent to the symbol E a .
In like manner we may write e AD 0(ae, y}≈φ(x + h, y).
If now we suppose both sides operated on by the symbol e lD ', we have e H>'. e AD ι y) = e k0 'φ(x + h, f) = φ{x + h, y+k).
Hence φ(x+h, y+k)≈e κτ * +l arφ( x , y)
= {l + (AD + ⅛D') + ⅜(AD + w) 2 +. . .}φ(x,y) ≈φ(x,y) + h^- + Nh^⅛ + 2hk^⅛- +⅛ x ) + fec∙
dx dy 2 ∖ dx 2 dxdy dy 2 ∕
(Compare § 57; also Arbogast, Cal. des Dir., pp. 343-352.)
85. Another proof, by the method of operations, of the foregoing symbolic expression for Taylor’s theorem may be added.
It has already been shown that when h is infinitely small we may write φ(x + h)- φ(x)—hφ'(x') = hDφ(x}, . ’ . φ(x + h) = (1 + hD^)φ(x).
In like manner
(l + ΛD) 2 ≠(it!) = (l + AD) φ(x + h} = φ(x + 2h ∖ &c.
And in general φ(x + rih) = (1 + hT>y n φ(x).
Now suppose nh≈a, and we get α φ(x + a) = (l+hD) h φ(x).
But when h is infinitely small, we may, by analogy (see§ 21), assume 1 a {l + hT)) h u=e O u, .∙. (l + ⅛D)*u-e αl, u.
Hence φ{x + a)= c a °φ{x') , as before.
86. Again, as in § 84, representing the symbol e υ by E, we may write eWE∕(≈)∙
Also, if Δ prefixed to any function of x denote the operation of taking the increment of that function when x receives the increment unity, we have
∕(z + l)-√(z) = ∆∕(x).
Accordingly E∕(x) ≈= ( I + Δ) f{x).
And, by the index law, we have
E"∕(z)-(l+∆)"∕(a r), or ∕(x + π) = ^l + n∆+ — ^ t g 1 ^∆ 2 + . . .^f{x)
-/(«) + τι∆∕(x) + 7 L∙ δ 2 ∕(z) + &c-
Adopting the notation ≠(ar) = u x , φ{x + h)≈ u x - ∖ -k, &c., this leads to the following fundamental theorem of the calculus of finite differences u x -i-n=ux+n∆u x + ~- Δ 5 ⅛+ . . .
1 · Δ
Again, since
Δ = E-1, we have Δ n u x = (E-1)"«».
Hence, in like manner,
Δ n M x -= w l+n -nUχ +n - 1 + ~ 1 - u x+b . 2 +... + (-l)*u i .
1 · Z
For example,
∆ n z m =(z + τι)∙*-n(ic + n-1)* ’ +.. . + (-l)"ai ,,i .
Again, if ∆"0 m represent the value of ∆ n x m when x=0, we have ,
Δ"0"=n*-n(w-l)∙+⅛J⅛∖n-2)- - . . . +(-l)"~⅛.
Jt. z
The numbers represented by the symbol Δ n 0 m , called the differences of the powers of zero, are of frequent occurrence in analysis, and their values can be readily tabulated from this series.
87. Again, since
W = * ⅛ u + v= (D 1 + D a )uu, in which we suppose D 1 to operate on u only, and D a on υ only, we infer that T> n {uv) = (D 1 + D a )" uv
= ( Df+nD 1 "-iD a +⅛≤ j D 1 "- 2 D 2 + . . . I uv d n u dv d n ^ 1 u n(π-l) d 2 v d n ^ 2 u d n υ ^ v dx n + n dx dj^ τ ι + 1.2 dx 2 + * ' - + u dxP
This is Leibnitz’s theorem, given in § 27.
This result can be extended to the πth differential of the product of any number of functions.
88. More generally, if ψ(z) represent any function of x, and if f{x) be any rational function, and we suppose D 1 operates on u only, and D a on ψ(ic) only, we have
∕(D)ψ(i r>=y(D 1 + D a )≠(i r)u
= {∕(D 1 ) + D a ∕'(D 1 ) + . . . }≠(z)u
= ψ(3 !)∕(D)tt + ψ'(x)∕(D> + ⅛⅛∕"(D> + &c.
1. z
In like manner the equation f{x)mu = φ{D)f{x)u-φXD)f{x) u + ^f f {x)u- . . .
1 · z can be established.
These expansions form the basis of Hargreave’s well-known memoir on the ‘ ‘ Solution of Differential Equations” {Philosophical Transactions, 1848). Hargreave observes that on mere inspection of these results it is apparent that if D be substituted for X, and - x for D, the former equation transforms into the latter. Hence, in any differential equation and in its symbolical solution, if the foregoing substitutions be made we shall obtain another form, accompanied with its symbolical solution. This principle was applied by Hargreave to the solution of several classes of differential equations.
89. Again, if in Leibnitz’s theorem we make v=e ax , we get
D"(c αx u)= e ax ^ D" u + nal) n * l u + —γ~α 2 D ’ * ^ 2u + · · ·
= e ^D"+nαD"-ι + i L^∑2 α iD n - 2 + . . .)u
-C κ (D+α)"w.
Accordingly
(D + a) n u - e ~ ∙ x D"(e ox u).
Hence we readily infer that, if f{a) represent any function involving only positive integral powers of a, we shall have
∕(D + a)u<=e~ ax f{T>')e ax u.
Again, if this be transformed by assuming e x -y, we have ~ = y, (λΛ and (⅞)"-(≈'⅛) w -*' d ' m ∙
Hence the foregoing result may be exhibited as follows :—
∕(yD' + d)u = y - a ∕{yT>')y a u .
This may be written
∕(arD + ai)u =x ^ α ∕(zD)z β u.
90. The interpretation of negative and fractional powers of a symbol of operation is a subject necessarily suggested by the introduction of such symbols. We pass over all allusion to the case ol fractional powers, as no satisfactory theory for their interpretation has as yet been arrived at. The interpretation of an integeι negative power of a symbol is easily established, and is in all cases of the nature of an inverse problem.
For instance let τ be a symbol of operation such that TΓU=V , then, if v be given and u unknown, we may write u=τr~ 1 υ, and the problem contained in the inverse symbol of operation wil be answered when, by any process, we have determined u so as t( satisfy the equation ru=v, or ιrιr~ l v=v. In other Words, w< define the inverse symbol π^ 1 to be that which the direct operatior τr simply annuls; and this is in accordance with the analogy o: ordinary algebra.
For example, since ^Df{x)=f{x), we write ^D^ 1 f{x')=f{x) , anc the symbol D^ 1 is equivalent to an integration. In like manne: D^" is equivalent to n successive integrations.
Similarly the symbol (D + α)^" is regarded as the inverse of tin symbol (D + α)", ι.e., such that
(D + α)" (D + α)^"u = tt.
We now proceed to investigate how far the equation
∕(D + α)u= e - ax f{D)e ax u holds for inverse symbols.
We have already seen that when n is a positive integer
(D + α)"u=e^ αx D n e αx ω = v, suppose; .'. w = (D+a)^"v.
Moreover from the equation e -axj)n e ax u - υ we get D n c ax u=e ax υ, or u≈e^ ax Ι)- n e ax v.
Consequently (D + a) ^ n v = e ~ a *D ^ n e ax v,
Hence we infer that the symbolic equation also holds for negativ< powers of D.
91. In general, since
D.e≠<¾ = β ≠(*){D + φ'{x)}u, we have ∣D + φ'{x) ∣ u = e - .
Again
]D + ≠'(≈) , and in general
{D + φ ’ {x) y u=e~^ι> n e^u , where n is an integer.
From this we conclude that in all interpretable cases we have
∕{ D + ≠'(z) }m= c -≠ w ∕(D) c ≠<⅜ .
The results here given have been generalized and extensively em ployed in the integration of differential equations by Boole. Set Philosophical Transactions, 1844; also Boole’s Differential Equa tions, chapter xvii.
92. We conclude this short account of symbolic methods bj applying them to establish one or two well-known formulae.
It has been shown already (§ 84) that we may write
(e*» - l)≠(ar) — φ{x + h)- φ{x)
Hence φ{x) = (c ad -1) - 1 {≠(x + Ä) - φ{x)} .
Multiplying by lι, and operating on both sides with the symbol of differentiation D, we get j ≠(x+⅛)-≠(x) J .
But, by analogy from § 53, W⅛' may write (⅛X≠< x+λ >-≠ w ) =0-'⅞÷ b ^-SS÷∙∙∙X*÷ λ >→ w )
= ≠(≈ + Ä) “ ≠{aτ) —+ Λ) - φ'(x) J + yA ) φ"(x + ∣ <')- ≠''(jr)) - τ ¾-. i ≠* , <≈≈+*>-≠ 1 '(≈') i ≠ ⅛⅛∙
1. Δ. o . 4 ( )
Hence φ{f + h) - φ(x) + ~ j≠'(i i + Λ) + ≠'(*)∣ - § ∣≠"(⅛ 4 Λ) - φ , '{^ h* i y + y⅛o γ≠ iv (aj + λ ->- ≠ lv (⅛)j -' · ' ·
+(j - 1JΓ ⅜^ .<f"¾⅝ ∙^≠ 2 "( aj + λ ) - ≠ 2n ( aj ) ∣ + &«·
This result is due to Stirling, and häs important applications.
To complete this proof it is necessary to consider the question of the convergency or divergency of this series. On this investigation see Bertrand, Calcul Intégral, Art. 374.
93. Again, in the calculus of finite differences, if we consider the finite symbol of summation 2 as the inverse to that Of finite differences Δ, we have
2≠(aj) = Δ -1 ≠(a!) = ι≠(ic )
-( d ' , - 4+ i⅞- 1 ⅛÷∙∙∙>>
^∕≠(∞)^-⅛≠(^) + A≠>)- 1 -⅛- i ≠" ’ (⅛) + &m J 1 . Δ 1 · Δ i O. 4
-c +∕≠f*-⅛ i ÷⅛≠'w-⅛<w+30k≠' w · · ·
This theorem is due to Euler; the foregoing demonstration was given by Gregory {CaVnb. Math. Journal, 1837).
On the limits of the remainder after n terms in this series, see Boole’s Finite Differences, pp. 91-93; also Mr J. W. L. Glaisħer, in Quarterly Journal of Mathematics, 1872»
In concluding this brief account of symbolic methods we may observe that the general principles of the theory øf operations have been studied in a comprehensive manner by Grassmann, and by Hankel, who applied them to the general theory of complex τaτh ables and of quaternions. See Grassmann’s Ausdehnungslehre (1862), and HankeΓs Vorlesungen über die Complexes Zahlen, 1867. The reader will find Grassmann’S method fully discussed in HσtieΓS Calcul Infinitésimal, vol. i.
We add a few miscellaneous examples of these methods;
(1) Prove the symbolic equation
α∙"D'¼4 = arD(xD-l)(ad)-2) . . . (xD - n + l)u.
(2) Prove that
D∙∙(e β ¾") = ^y^ r D n (e"α*).
(3) Prove the symbolic equation in finite differences
(E - α) n X = α n +≈∆ n α- n X, where E is the symbol e D (Gregory, Camb. Math, four:, 1837).
(4) If τr and ρ be symbols of operation such that
πp-pτr = p 1 , πp 1 -p 1 π≈=p. 2 , ιrp a - p 2 r = p 3 . .
prove the following symbolic equation
∕(fi∕>=p∕W+p 1 ∕'(r)+⅛'(τ) + . , .
1. Δ
See Donkin, Camb, and Dub. Math. Jour., 1850.
(5) From the preceding thé following symbolic équations can be readily deduced.
∕(d + X')-∕(D) + X>(D) +i ¾ ∣> f D) K . ..
f( x + ⅛PΛ . f71 >≠XP) , Ae) £W j ∖ ≠(D) ∕-∙A*) +∕(a ≈)≠(D) + 1.2 ≠(DT ,,
(Donkin, Hid.)
(6) Every differential equation of thé fofm
∙i(α + iz + cχ- 2 + . . ,)D ft + (α' + δ⅛ + . . . )D"-ι÷. . .∣u = X can be ttànsfolmed into the shape
{∕ 0 (≈D) +∕ 1 (icD> +∕ 2 {xD)≈ 2 + . . ,] u * X. (Boole. )
(7) Apply the method of Operations to the proof of HerSclieΓs expansion for F(e i ) {Philosophical Transactions, 181 δ).
If we make 0=0 in the equation f{tJ)^ =d e f{l), we have
∕(0=∕(R) c ° i > where D represents ihe symbol £ .
αθ
If now ∕(Z) = F(e i ), we get
F(e‘) = F(e D ).e« = F(l + A)e^
= F(l + ∆)∣l + 0i + ^ + i .. (
= F(l) + iF(l + ∆).0 + 2iF(l + Δ).0 2 + &t.
1 · Δ
(Gregory, Camb. Math. Jour., 1838.)
(8) Prove the equation f{D)φ{e-)e^≈φ{Ε 1 )f{^, d — where D represents^-, and E represents e d <r. ais ° z + r W = ≠(≠+ x ∖m · ∖ ax J ∖ ar J
(Bronwin, Camb, and Duh Math. Jour., 1848.)
(9) Prove the symbolic equation
≠^)∕(D)X=∕(D)≠(z-i)X, where x is to be regarded as a variable independent of the operation D, but which, after the operations, is to be replaced by x. (Crofton, Quar. Math. Jour., 1879; also Donkin, Camb, and Dub. Math. Jour., 1850.)
Change of Independent Variable.
94. In the application of the cälcülüs it is often neces⅛afy to adopt in our equations hew independent Variables instead of those originally selected. ‘
Thus, suppose it be required to transform à function of y, x, -f, , &c., into a function of y,i, -f-, , x being supposed a dx dx 1 ’ dt ’ dt i oil function of t.
Let the functions ⅛ É-?, &c., bθ represented by ff, d", &c., (jLt (Uj then we have iħ all cases du _ 1 du . dy dy , dx J ft ’ dx aJ ’ dt ’ also ⅛ <√l⅜∆ 1 d ( 1 dy ∖ fi. f <ffy ^y ∖ dx 2 dx ∖ vd dt ) U dt ∖ x' dt J ap*∖. dt 2 dt ) ( sc ' _ a ∙"⅛ ∖ I x ' tf V _ s fw ∖
X . d 2 y d{ θt 2 dt] 1 d( dtT ciï I Again —∕ *v dt ∖ and so on for differentials of higher order.
If y be taken as the independent variable, we have
⅛-l, ⅛-0,4c, dt at 2 ·
Hence dx dx ’ dx 2 ~ /^’dy 2 ' dy ∖ dy) d≡*°j⅛y I 3 ∖ df) ~Ty^dy^ ∖ , itc ∙
The formula for the change of thé independent Variable Were giveiι for the first time in the Traité des infiniment petites of L’Hôpital. The geħeral theory of transformation Was discussed at Considerable extent by Euler in his Calc. Diff.
In the ease Of two independent vafiables i suppose wè äfe gi⅞h x=φ{r, θ), y=ψ(r,θ). r r1 dv dυ dx dv dy . ^r dx dr + dy dr ’ dv dv dx^dv dy dθ dxdθaylι∂' dv dy dv dy dv dθ dr dr dθ liencθ dx~dxdf_dxdy 5 d& dr dr dθ dv dx dv dx dv d θ dr dr dθ dy dx dy dy dx dr dθ dr dθ
In like manner ¾ , ¾ can be deduced, but their general values dx 2 df t are too complicated for insertion here.
A case which comftιonly arises is in the transformation from rectangular to polar Coordinates.
In this case, we have α=r cos θ, y=r sin θ , and dv jiv sin 0 dv
- r - =⅛ COS θ~ 1 wτ »
dx dr r dθ dv , .dv , cos θ dv
- j - = sin 0- 5 -÷- -,. ·
dy dr r dθ ιr d' 2 υ ( ,d sin 0 d ∖ f Rv sin θ dv ∖ tienee 35 )(∞ a ⅛- —
. Δ ά 2 ν 2 sin 0 cos 0/ 1 dv d' i v ∖ " cos ⅛ + ~7 ∏ di-≈i) sin 2 0 dv sin 2 0 d 2 v + f dr r 2 <Z0 2 *
The corresponding value of ~ a is got by substituting - 0 instead - ay' i a of 0 in the last equation. Hence wα easily find d*v d⅛ d t v 1 dv 1 dfv clx i + dy 2 ~ dr 2 * r drf 2 dθ 2 '
95. Another important case, which is of extensive application in geometry, is that of linear transformations. ,
Let us consider the case of three variables, and suppose x=aX + bΥ + cZ, y=∙a'X + bH+c'Z, z=α"X + 5"Y + c"Z , . 1 dv dv , , dv ,,dv then = a ± a ' + α " , dX dx dy dz dv .dv 1 ,,dv . ...dv dt~ i TV i TV t 1i∙ dv dv , dv ,,dv dZ^ c di +e dij +cr d~z∙ d 2 v f d . d l ,,d ∖ 2 -⅞vS=∙=(a - r + a'- r + a "- r I v dX- ∖ dx dy dz J JPv n , d 2 v n . „ d?v , λ , ,, cPv = α a - r -⅛ + 2aa - r - r + 2αα"- r - 7 - + 2α a '- r - r - dx i dxdy dxdz dydz , >∙id l v t l ,fl 2 v + a dy 2 * a d?·
Again, if we suppose ae', y', √ to be transformed by a similar substitution, i.e., of=aX' + δY' + iZ', f = a'X! + δ'Y' + c'Z ’ , &c., then, if any function u=φ{x, y, z) transform into ≠ 1 (X, Y, Z), we shall have φ(x+ksd i y + kf, z+kz') ≈ φ 1 (X + kX', Y+⅛Y', Z + kZ , ).
If these be expanded, and like powers of k at both sides be equated, we have
4 +v⅛ +* ’ 4U = (χ '⅛+ γ 'Λ+ z ' u 1 ∖ dx s, dy dz} ∖ dX dx dZJ ( . d .d .d ∖ - ( v ,d - vf d 1 d ∖ a .
(⅛ + ⅛ + ⅛) tt ~(⅛ + ⅛ + ⅛) u ' &c ·
Consequently the functions sc'~ + √⅜- t ÷ z'⅜ i , &c., are unaltered √ dx a dy dz by linear transformation. These functions have important geometrical relations with the original function. Many applications of these principles will be found in Salmon’s Higher Plane Curves, as also in his Geometry of Three Dimensions. A few additional examples are added for illustration.
(1) If ae=tan 0, <Fy, ⅛ ⅜ t y =n dae a l + ≈ a <⅛c (l+ic 2 ) a transforms into
⅛ + ≈'^ 0 ∙
(2) If z be a function of x and y, and u=px + qy-z, prove that when p and q aretaken as independent variables we have du du dPu t d 2 u -s d 2 u τ dρ"= x, dq dp i "f't~s 1, dpdq^rt^s i, ' clq ι ^°rt^s 2 ’ where p, q, r, s, t denote the partial differential cOéfÔCiéfitS öf z with respect to x and y, of the first and second orders.
(3) In the linear transformations in § 95 the determinant (αδ'c") is called the modulus of transformation, and the transformation is said to be orthogonal when
* 2 + 2∕ 2 + z a ≡X a + Y 2 ÷Z 2 .
In this case the determinant [table]
is unaltered by the transformation.
Jacobians.
96. We now proceed to a short treatment of a remarkable class of determinants first studied by Jacobi {De determinantibus functionalibus, Crelle, 1841), iιi developing important generalizations of the fundamental principles of the differential and integral calculus.
If u 1 , u 2 , u 3 , . . . u n be functions of n independent variables aei, x 2 , x 3 . . . x n , then the following determinant [table]
was called Afunctional determinant by Jacobi. Such determinants are now more usually known as Jacobians, a designation introduced by Professor Sylvester, who largely developed their properties, and gave numerous applications of them in higher algebra, as also in curves and surfaces.
The preceding determinant is frequently represented by the abridged nôtation d(u l , w 8 . . . ⅛ n )
d{x x , x 3 . . . x λ )^
The following discussion, for brevity, is limited for the most part to the case of three variables, but it Can be rëàdily extended to any number.
97. Altering the notation, we suppose it, v, w to represent functions of three independent variables, x, y, z; then (Bertrand i Lionville's Journal, 1851), if we attribute to each variable an infinitely small increment, there will result »Corresponding increment for each of the functions. If UoW we choose arbitrarily a number of different systems of ificreménts, there will result a corresponding number of systems of increments for the functions.
Accordingly, representing the increments of x by d 1 x, d.pe, d 3 x , and similarly for the other variables, we shall have [table]
Consequently, by the fundamental rule for the multiplication of determinants, we shall have [table]
Let the first déterminant be rdpreàented by (A), the second by (B), and the third, of Jaxobidii, by J, and we get J = A That is to say, the Jacobian is the ratio of the determinant of the system of infinitesimal increments of the functions to that of the increments of the variables.
This may be regarded as a generalization of the definition of the derived function in the case of a single variable.
98. Again, when the fuηciibns u, v, w are connected by any relation their Jacobian vanishes.
For suppose u, v, zo to be connected by an equation
F(w, v, w) = Q, for all values of x, y, z ; then, since in this case
£_ 0 , ∣!. 0 , dx dy dz we have dF du dF dv_ ι dF dw du dχ+ dυ dx dw dx ’ dF du dF dυ ι dF dw~θ du dy + dv dy dw dy ’ dF dw ψ dF ⅛ + <fF d^ =0 du dz dv dz dw dz
Consequently, eliminating dF dF dF du ’ dv ’ dw , we eet [table]
This is an extension of the theorem that when a function of a single variable is constant its derived function is zero.
The converse of the preceding theorem can be established, viz., if J = 0, then the functions u, v i w are no longer independent of each other.
These results are readily extended to any number of variables; thus, whenever the functions u 1 , w 2 , . . . u n are connected by a relation, lt > ’ · · ’.⅛) = Q; and conversely.
d(x 1 , X i , . . . X n )
99. Again, if u, v, w, instead of being given explicitly in terms of x, y, z, be given implicitly, i.e., if they are connected-with them by three equations of the form
F 1 (x,y,z, u,v,w) = 0, F 2 (x,y,z,u,v t w) = 0, F 3 (ae, y,z,u,v,w)≈0, we have, adopting the same notation as before, dF 1 7 dF 1 j dF 1 j dF 1 , dF 1 . ι dF 1 , n - r 1 d l x + - τ j d 1 y + - 7 1 d 1 z + - r 1 d 1 u+ - r i d 1 v + -÷fd 1 w=0, dx 1 dy lt, dz 1 du 1 dv 1 dw or dF, j dF lj dF 1 , dF 1 , dF 1 , dF 1 .
-~ d 1 x H—j- i d 1 y d— r i d 1 z = —-,~ i d 1 u — -H d 1 v —τd 1 w, dx 1 dy ιy dz 1 du 1 dv 1 dw and similar equations for the increments d 2 x. . . d 3 x, &c. . . . , as also others derived from the functions F 2 , F 3 . . . Hence, as before, [table]
This result, when generalized, may be written as follows :— [table]
100. We shall next consider the generalization of the elementary theorem *f.
dx du dx
If we suppose ≠ 1 , ≠ 2 , ≠ 3 to represent functions of u, v, w, while u, v, w are functions of x, y, z, then, adopting the same notation as before, and representing the determinant [table]
by C, we have by § 97 [table]
and similarly [table]
Hence [table]
Consequently, the Jacobian of φ 1 , φ 2 , φ 3 with respect to x, y, z is equal to their Jacobian with respect to u, v, w multiplied by the Jacobian of u, v, w with respect to x, y, z.
This is the required generalization in the case of three variables.
101. Again, if u, v, w be functions of ae, y, z, we may regard x, y, z as functions of u, v, w ; and it follows immediately that the Jacobian of w, v, w with respect to x, y, z is the reciprocal of the Jacobian of x, y, z with respect to u, v, w∙, i.e. [table]
This, when extended to n variables, is the generalization of the theorem that the derived function of y with respect to x is the inverse of that of x with respect to y.
The preceding demonstrations are readily extended to any number of variables. When generalized for n variables, the results are written in abridged notation thus—
d(φ 1 > ≠a > · · ≠*) tm rf (≠ι , ⅛ ∣ · · ⅜) χ d( l u 1 , ⅜⅛ , . . . ⅜⅛) d(x 1 , x a , . . . a⅛) d(u 1 , u 2 , . . . u n ) d(x 1 , x 2 , . . . x n ) ’ and rf(¾ 1 , ⅝ ,. . . ‰) χ d(x 1 , a⅛ , . . . a⅛)
d(ir 1 , x 2 , . . . x n ) d(u 1 , u 2 , . . . u n )
102. Again, the Jacobian of any system can be expressed as a monomial. This result can be established as follows
Reverting to our original discussion, it is readily seen that of 2n quantities, αq , x 2 , x 3 . . . x n , u 1 , u 2 , u 3 , . . . u n , connected by n equations, when any n are chosen at pleasure the others are capable of determination. Consequently, if n-1 of them be supposed to remain invariable, all the others may be regarded as changing simultaneously, and the ratios of their infinitely small increments are determined. Hence we may suppose our n systems of simultaneous increments attributed as in the following table :— [table]
The first line indicates that the first system of increments attributed to 0! 1 , x 2 . . . x n are such that u 2 , u 3 . . . u n do not change; in the second line we suppose that increments of the second system are such that a! 1 , u 3 . . . u n do not change; and so on.
Again, since for these values the determinants A, B, reduce to their diagonal terms, we have, in this case, by § 97, j dju^ d 2 u 2 d 3 u 3 dnun d 1 x 1 ' d 2 x 2 ’ d 3 a¾ ' ’ ’ d il x n '
Also, by what has been stated above, the ratios d∙γUγ d 2 u 2 d 3 u 3 dnun . dγrγ d 2 r 2 d 3 X 3 dnXn can each be determined in this case from the given equations.
Consequently the Jacobian of our system is the continued product of ⅛ ⅛ du s dx l ’ dx 2 dx n
In order to calculate it is necessary to express u 1 as a function of x 1 , u 2 , . . . u n ; and similarly for u 2 , u 3 , &c.
103. For example, let it be required to find the Jacobian of the system
O! 1 = r cos θ l , x 2 =r sin θ 1 cos 0 2 , x 3 = r sin 0j sin 0 2 cos 0 3 , x n -r=r sin 0 1 sin θ, 2 . . . cos θ n -ι, x, l =r sin 0 1 sin 0 2 . . . sin 0 i ,-ι∙.
Here, squaring and adding, we get x 1 2 + a⅛ 2 + . . .+α⅛ 2 =r a .
We shall employ this instead of the last equation of the system.
Hence, adopting the conditions laid down in § 102, we get dx, . . dx 9 . λ . λ q dx n r s - 1 = - r sin 0 1 , - v j =∙ - f sin 0, sin 0.,, . . . &c., —-- = - . dβ 1 1, dθ 2 ^ dr x λ
Accordingly, the Jacobian of the system is
( _ 1 ∖ b - 1 r ** sin n ~ 1 0 1 sin ,, ~ 2 0 a . . . sin0, 1 - x aen
= ( - I)" -1 r"~ 1 sin"^ 2 0 1 sin n - 3 0 2 . . . sin 0, 1 -2
104. Again, suppose u 1 , u 2 , - . . u n to be the partial derived functions of a given function of the variables x 1 , x 2 , . . . x n ; i∙e., let df df df uι≈f- , u, 2 = f- , . . . ‰ = √ . dx l dx 2 dx n
The Jacobian becomes [table]
Such a determinant is called the Hessian of the function f(x l , x 2 , . . . x n ), after Hesse, who first introduced such determinants into analysis, applying them in many investigations of fundamental importance in the theory of curves and surfaces.
105. Again, in the Jacobian d(Vι, y- i ∙ . y,>)
<Z(aj 1 , ¾ . . . x h ) ’ if the functions y 1 , y 2 , . . . are fractions with the same denominator, i.e., such that u, u<>. . un yV∙-J> · · · U∕ Its IL we have „ dm du, du u ~j≈ u j ~ u ι ~τ~ ’ dxk dx ∣ c i dxk n dy 9 du 9 du uΛ= u J~ u ∙2 ^T~ » dx ∣ c (Lx ∣ e dxk
Hence [table]
From this, by elementary properties of determinants, we get [table]
Hence [table]
This latter determinant has been denoted by K(w, u 1 , . . . u tl ). It possesses interesting properties. For example, if u,u 1 , . . . u n are connected by any homogeneous relation, then
K(w, u 1 , ... u n ) - 0.
This follows from § 98, since the quantities y 1 , y 2 , '. . . y n are in this case connected by an equation.
It is seen without difficulty that Jacobians and Hessians are covariants. That is, if the functions be transformed by linear substitution (§ 95), the Jacobian of the transformed functions is equal to the original Jacobian multiplied by the modulus of transformation; and similarly the Hessian of the transformed function is equal to that of the original function multiplied by the square of the modulus. It can also be seen that, when the transformation is orthogonal, the Jacobian and Hessian are unaltered by the transformation.
PART II.
Integral Calculus.
106. The integral calculus may be said to have taken its origin from the "methods employed by Cavalieri, Wallis, and others, for the determination of the quadrature of curves and. the cubature of surfaces. These methods, as we have seen, consisted in the division of the required area, or volume, into an indefinite number of thin slices, or “elements”; and then from the law connecting their successive values the sum of all the elements was determined—or rather the ‘ ‘limit” to which that sum approached when the number of elements was indefinitely increased. The processes thus employed were developed and reduced to a suitable notation by Newton and Leibnitz. Thus, adopting the more modern nomenclature, if φ(x) be a function of x which is finite for all values of x between the limits a¾ and X, and if we suppose the interval X - x 0 divided into n parts, x 1 -x 0 , x 3 -x 1 , a⅛-a⅛, . . . X-ic n -ι, then, multiplying each element by the corresponding value of the function, i.e., x 1 -x 0 by ≠(⅞), &c., the sum
, S = (x 1 - a⅛)≠(⅜) + (a⅛ - x l )φ(x 1 ) ... + (X - zn-i)≠(icn-ι) has, by elementary algebra, a finite value,, which may be represented by (X - ⅜)≠(ξ), where ≠(ξ) lies between the greatest and the least value φ(x) admits of between the limits.
If, now, we suppose the number of elements increased beyond limit, so that x 1 - x 0 , x 2 - x 1 , &c., may be regarded as each becoming indefinitely small, then ultimately the value of S attains to a certain limit, which depends only on the form of the function φ(x), and on the extreme values X and x 0 . In this stage, introducing the symbol of integration Z and adopting the notation ∕ φ(x)dx, instead J Jχt of S, we write J x φ(x')dx=lim. [(a⅛ - a∙ 0 )≠(a⅛) + (x a - x 1 }φ(x 1 ) + . . .
+ (X-z n -ι)≠(‰- 1 )]-(X-a∙ 0 )≠ ⅛ + 0(X-α! o )∣- , in which 0 lies between 0 and 1.
For greater simplicity, it is usual to suppose that the increments x 1 - x 0 , x a - x 1 . . . X - x n -1 are all equal. In this case their common value h is equal to the fraction — — 0 ; and S becomes n h {>o) +Λ⅞ + Ä) +∕(⅜ + 2Ä) + . . . +∕(X - 2Ä) +∕(X - Ä)} .
Again f -∣a⅞ + 0 (X - a⅛)∣∙ represents the mean value of ∕(ar), as x proceeds by equal infinitesimal increments from the value x 0 to X. The application of the integral calculus to the. solution of questions on mean or average values is founded on the result here given. Thus, denoting the mean value of φ{x), between the limits X and ∙τ 0 , by M≠(z), we have
1 ∕ ,χ M≠(x)= v - ∕ φ{x)dx.
X ¾*'∙ r o
107. If in the definite integral f φ{x)dx the upper limit X be Jχo conceived to vary, x 0 remaining constant, the integral itself will vary; and if we replace X, regarded as variable, by x, the integral may be regarded as a new function, F(ae), of x, determined by the equation
F(ae) = ∕^ ‘φ {x)dx≈{x-xj) φ [a⅛ + θ(x -ir 0 )J ·
This function vanishes when x= a∙ 0 ; . ∙. F(a ’ 0 ) ∙= 0.
Also, by the differential calculus (§ 46) we have
F(«) = (x - a¾)F'[a∙ 0 + 0(x - α⅛)] .
Consequently ≠[⅜ + θ(x - a⅛)] = F'[ar 0 + θ(x - a⅛)]∙
Again, making x=-x 0 , we get
≠(z∙ 0 ) = F'(a∙ 0 ); and, since x 0 may have any value, we have in general
≠(x) = F(x).
Accordingly we may write
∕ x F(xχ ic =F(X)-F(⅞).
J x ⅛
Hence the process of integration is reduced to the determination of a function F(x) when its derived function F(ae) is known.
We shall illustrate these preliminary remarks by one or two examples.
Ex. 1. Find the limit of the sum of the series n η n n 2 +l 2 % 2 + 2 a ' ' ' + 2n 2 ’
when n is indefinitely increased.
Let dx= — , and the limit of the series is easily seen to be reiz.
presented by ∕" 1 dx . ιr . d l , 1
f 0 Ι + ae 2 ’ 4 dx v ' 1 + a: 2 Ex. 2. Find the limit of the sum
L- + --X^ + -l a- 1 √λ 2 -1 2 √τι 2 -2 2 √n a -3 2 √π 2 -(n-l) a ’
when n is indefinitely increased.
Here the required limit = f -Af. ~∙ =f- , JO √l-z 2 2
108. We might have started from the preceding result as the definition of the integral calculus, and regarded this calculus as the inverse of the differential. Thus, as in the differential calculus we investigate the rules for proceeding from any primitive function F(aj) to its derived function F'(aj), so in the integral calculus our object is the converse, viz., to determine F(ae) when F'(x) is given; or, in the language of Newton, “to find the fluent of a given fluxion.”
It may be here remarked that it has been shown from geometrical considerations, in § 23, that such a function always exists.
In the differential calculus rules have been laid down for the method of determining the differential of any function. There are, however, no direct rules for the inverse process, except by retracing the steps by which the derived has been deduced from the original function.
Accordingly, the integral calculus is based on the differential, and to each result in the differential calculus corresponds another in the integral. Moreover, as F(x) and F(aj) + C (where C is any arbitrary quantity that does not vary with a?) have the same differential, it follows that to find the general integral of F'(z)dz we must add an arbitrary constant to F(ic).
109. The following elementary integrals (omitting arbitrary constants) are easily arrived at, and are called fundamental integrals, to which all others that admit of integration in finite terms are ultimately reducible—excluding higher transcendental functions :—
r x n dx= ^ff J d J≈∖ogX, J n+1 J X J ∖ vcLχdx= - cos x , ~∕c03χdx=smx, ∕ dx . f dx . cos⅛ y sιn⅞
√ r dx . , x Γ dx . s , -7-5 ; = sin ^ 1 — , /—==≈=- = log (ae + √α 2 + x i ),
√α 2 -αf 2 « y√a 2 + x 2 6 √ C dx 1 , , X f a x
-5-—⅛=- tan^ 1 --, ∕a x dx≈-. . a 2 +x s a a J log a
110. A number of integrals can readily be reduced to one or other of the above forms. A few elementary cases, such as frequently occur in practice, are here given. We commence with the integral
(1) ∕ dx . . .
J(x-d)(x-β)
Here 1 = 1 ∕ 1 1 ∖
(x-a)(x-β) a-β ∖ x-a x-βj' √ r dx 1 , X-a lo°* . (x-a)(x-β) a —β o x-β
(2) More generally, the integral ∕* dx J a + 2bx+cx j may be written in the form √ r cdx
(caj + δ) a +αc-ft 2 ’
or, substituting z for ex + b, Γ dz Jz-+ac-tf i
The form of this integral depends on the sign of ac - b 3 .
If ac-l? >0, we have ∕ dx 1 . , cx+b
∏ J= - -7 tan -1 —. .
a + 2bx + cx i √αc-δ 2 √αc-ft 2
If αc-Z> 2 =0, f^ dx -1
J a + 2bx+cx 3 cx + b’
If ac - Z> 2 < 0, the integral comes under (1), and we have √ ’ ^ dx 1 cx + b-Jb' 2 -ac a + 2bx+cx i 2Jb 3 -ac o cx+b + Jb' 2 - ac
(3) Again, since l + mx m b + cx lc-mb 1
a + 2bx + cx 2 c a + 2bx+cx i c a+2bx+cx 2 we have ∕ ’ (j + mx')dx __ rn Γ (b + cx)dx j t- l c ~ mb f__dx__ a + 2bx + cx 2 c J a + 2bx + cx 2 c J a + 2bx + cx 2 ‘
The integral of - (δ + ca 0<% e ⅛ ι j 0 g r a q. 2δaι + cx 3 ), and the latter a + 2bx + cx 2 6 integral has been obtained in (2).
(4) Next, to find P- —— . y sin X cos X
Here Λ-÷∖- - = ∕L 1 - . dx _ = ∕'d(tanx) = log (tan x). J sin x cos x J tan x cos⅛ J tan x
In like manner, (5) ∕sħ^∕-4-⅛- 1 °≡(tm i)·
√ ≡n τ ∞s^-
Hence we get /2ÈL = i 0g , ∣ an ( TL + 2L A . Jcosx 6 ∖ 2 4 J
(6) Again J^ tan¾ dx∙=f sec⅛ dx-f dx^ tan x - x.
111. The number of independent fundamental formulae must ultimately be the same as the number of independent kinds of functions in analysis. The ordinary elementary functions may be briefly classed as follows :—(1) algebraic functions, powers and roots, such as x m , for fixed numerical values of m, &c. ; (2) trigonometrical functions, sin x, tan x, &c., and their inverse functions, circular functions, sin^⅛, tan^⅛, &c.; (3) exponentials a x , &LC. , and their inverse functions, logarithms.
Several other transcendental functions have been introduced into analysis, such as elliptic and hyper-elliptic functions, gammafunctions, and others. We propose subsequently to give a short account of the elementary properties of some of these functions.
112. The reduction of an integration to one or more of the preceding elementary forms is usually effected by one or other of the following methods :—(1) transformation to a new variable; (2) integration by parts; (3) decomposition into partial fractions; (4) successive reduction; (5) rationalization. Examples of these methods will appear in succeeding paragraphs.
113. The method of integration by substitution corresponds to a change of the independent variable. We shall exemplify it by a few simple cases.
∕* . · · ·
Ex. 1. Let u = ∕ z V— , m being a positive integer, y (α + bx) n
Assume a+bx≈z, and the integral transforms into
1 ∕ ^(g ~ a) m dz hm+l J jjn.
If (z - df n be expanded by the binomial theorem, each term can be separately integrated.
y r dx
(ae + àjJ a + 2bx+ci?
Let x + a=-⅛- , and the integral transforms into Γ dz J Ja ’ + 2b ’ z + c ’ z 2 ’ where α'=c, b ’ = b - ca, d≈a- 2ba + ca 2 .
„ f dx LX ’ 3 ’ u Jla + cx^- Let ae= —, and we get z r - zdz _ 1 _ x U Jlaz 2 + c) i - alaz 2 + c)⅛ ala + cx 2 )⅛ _ p dx U Jl a + 2ta + ca, ,2 )⅛
Here w “/r— ⅞21f ic .⅞ ∖ 8)"⅝∙ J [ac- δ 2 + (c25 + δ) 2 p
Let cx + b=z, and it transforms into the preceding integral; hence U ≈±* . . lac - b i ) la+2bx + cx 2 V Ex. 5. u≈-∕^ i .
J (α' + c'z a )(α + cic 2 ) i
Let la+cx 2 ^ =xz, then dx dz
(a + cai 2 )⅛ - c→ 2 ’
and the integral transforms into ∕* dz J la!c - c'α) - α⅛ a ’
Ex. 6. « = /* dx
√√(α-φ-β)
Let x≈a sin 2 0 + β cos 2 θ, and we get
-⅞ a ,, Nla-x)lx-β) f . dx ≈20-2 sin^ 1 .
J *fla-x)lx-β) V a-β Ex. 7. «=/"— λ ∕ α "^ g ∖ .
J X 'X' x 2 -β 2
Here <^ 8 xdx x 'fla 2 - x 2 )lx 2 - β 2 ) √( β 2 -x i )lx i - β 2 ) ’
If we makex 2 =- in the former, and κ 2 =y in the latter, they each reduce to the preceding example.
Ex. 8. f.. . cfe . J a + b cos x
Here u= Γ- , J (a+b) cos a 4 + (α - V) sin 2 —
2 2
_ sec 2 — dx = ∕ 2 o ∕ - <⅛
J α + δ + (α - δ j tan 2^ Ja + b+la-b)S 2 ’
2
where 2=tan — .
2
This integral is a circular or a logarithmic function, according as a >, or, < b.
(1) Let a > b, and suppose δ≈≈α cos a, then we have u≈- Γ — —?— tan* 1 ftan — tan —) .
a J cos 2 — + s a sin 2 — αsιnβ ∖ 2 2 >
2 2
(2) If a< b, let a = b cos a, then
_ , , cos + sin — tan —
P dz 1 1 2 2 2
tt— / — = r—-. log - Z. J cos 2 4-2 2 sin’A 0sιllα cos-“_ sin — tan —
2 2 2 2 2
α — x 1 1 c ° s ~2~
"*r - =— log r^ ·
b sin a a + x cos - à
(3) If a≈b, the value of the integral is
— tan — . a 2
114. The substitution of an imaginary expression for a constant in an integral is often useful in evaluating integrals. For example, if in the equation
Äι α ⅛e≈-
J a we substitute α + iβ for a, it becomes ∕e aχ (cos βx + i sin fl⅛- e aa (cos ftr+t sin βx) J a + iβ
~ e g *(cos βx + i sin ßx^lg-iß) a 2 + β 2
Hence, equating the real and also the imaginary parts, we get 4*cosgsd≈^ g cos * te + * sin *0, √ a 2 + 0 2
∕≥≈ sin βx sin βx-βc 0 8βx) J a a + 0 2 omitting the arbitrary constants. These results can be easily verified.
115. The method of integration by parts depends on the following equation, which is deduced immediately from the relation dluv) = udv + vdu: — fudv = uv - fvdu .
Hence the determination of the former integral is reduced to that of the latter, and vice versa. Ex. 1. To find f tan -1 « dx.
Here J" .KΑ~ 1 xdx≈x^~' i x-J'-^-^=χ tan-⅛-⅜ log (l + α 2 ).
Ex. 2. Next, to find
∕c n log a: da;.
Let 1 a^ ’ + 1 m== log a;, v = -, n+1
then
4-log itl fc-≤≤ log≈-∕≤⅛-≤±i(log≈- ’ ∖.
J - n + 1 J n+1 n + l∖ n+lj Ex. 3. Again, to find
/log lx + ^fx i + a 2 )dx . J ∖ log lx + Va: 2 + a 2 )dx=x log lx +Vaj 2 + α 2 )
≈x log lx+ ^fx 2 +a 2 ) - Vz 2 + α 2 .
Ex. 4. /(log x) n dx . /log x) n dx≈ xllog x) n - »/(log x) n ~ 1 dx -=x(log x) n - nxllog x) n ~ 1 + nln - 1/log x) n ^ 2 dx.
Accordingly, by successive applications of this formula, the integral can be found whenever n is a positive integer. If n be a negative integer, the integral finally depends on f-^ ,x - , a form J lθg ae to be subsequently considered.
Ex. 5. ∕tf"(log x) n dx.
This is at once reduced to the preceding by making s=a^ n + 1 .
Ex. 6. fxT n e ax dx .
This is immediately reducible to Ex. 4 by making e ax ∙=z, It can also be deduced directly, since Γx^e ax dxJ ’ ^T~ fe ax tf n ~'dx. J a a J Ex. 7. Z⅛ e Z · Ans · τ~~ ·
√(l+a!) 2 1+a:
116. In general, if Q- u dnv du d ^' lv i d ^ u dn ^ 2υ 4-(-1 ∖ n d ^ υ dx n dx dx n ^ 1 ^dx 2 dx n ~ 2 ’ ' ' ' ' dx n we have e+( -1)"+> A≤⅛<i⅛.
J dx n + 1 J das n + 1
This result is readily proved by successive applications of the method of integration by parts, or can at once be verified by differentiation.
As an example, let us consider the integral
∕F(x)e⅛, where F(ic) represents a rational integer algebraic function of x, of the degree n.
Let w=ιF(z), ana , then t Γ + l‰ 0ι ' ,, a n + 1 ’ dz n + 1 ’
consequently we have ∕'e- x Ylx)dx≈ c - ∖ F(a≈)-≡ + . . . + (-i)-⅛Γ∣. cι L ® J This result can be also readily obtained by aid of the symbolic theorem of § 90, thus,
D^ 1 . e α *F(as) = e αz (D + α)^ 1 F(as)
-⅛*(1 + ⅞) F<z)
-√[ 1 -⅛ + ⅞-∙∙∙÷i- 1 ⅛>)∙
the remaining terms being neglected since D ,1 + 1 F(as) = 0. This result plainly coincides with that previously found.
More generally, if F(as, e ax , c bx ,. . e κχ ) represents a rational integer function of x, e ax , e bx , . . . the integral of
F(as, e ax , e bx , . . . e κx ')dx can be determined. For, this function, being composed of products of integer positive powers of x, c ax , &c., will consist of a number of terms of the form Ax m e ,ax c nbx . . . or Aas m d iβ + w6 + · · · P≈, each of which can be integrated by the preceding method.
Again the form ·
∕F(as, log as)das, is reducible to fF(e z , z)e x dz, by making x=c z , and consequently, when F represents an integer algebraic function, is integrable by the method considered above.
117. We next proceed to give a brief account of the treatment of the integral ∕2L^<Za,∙, in which f(x) and φ(x) are rational √ ≠(as) algebraic functions of x.
This class of integrals early engaged the attention of mathematicians. For example, Leibnitz and John Bernoulli, in the Acta Eruditorum (1702 and 1703), showed that such integrals depended on the method of partial fractions. The processes there given were simplified and generalized by Euler (Introductio in Analysin Infinitorum, 1748).
When the degree of f(x) is not less than that of φ(x), the expression =4~∖ can by division be reduced to an integer along with φ ∖ X) a fractional part; we may, therefore, suppose that we have reduced the degree of√(as) to less than that of φ(x~). Then, a being a simple root of φ(x} = 0, we may assume φ(x} = (x-a)χ(x), where χ(x) is not divisible by x - a.
If - we now make f(x) A f-fx) φ(x)~x-a χ(x) ’ we have Z⅛L A x (≈) + (as - α)/ λ (»), φ(x) (x - a)χ(x)
.∙. /(as) =» A x (as) + (a: - α}∕i(as).
This gives ∕ 1 (as) A⅜⅛). x — a
In order that the second member should be an integer expression, f(x) - Aχ(x) must be divisible by x - a ; hence we get
A = ≠⅛) = Z^) ,
^^ x (α) φ , (a)'
In like manner, if b be a second simple root of φ(x} = 0, and consequently a root of χ(x) = 0, we may make χ(x) ≈ (x - b)ψ(x). Hence we get
∕ 1 as B ι ∕ 2 (as) ; χ(x) x-b ψ(x) ’ from which it follows that b=M.
≠'(δ)
Finally if a, b, . . . l represent all the roots of ≠(as)≈O, no two of which are equal, we shall have
JL + B + ... + L , φ(x) x- a x-b x—l where A=⅞^ , B≡4% · · · ·
Φ («) Φ (⅛) Φ (0
Hence √ φ(x) φ(a) Φ(bι φ(l)
In the general case of multiple roots, we may suppose φ(x)≈(x-a) a (x-bY . . . (x-l) x , and assume ff)_ A 1 i a ∙2 + · A a φ(x)~x-a (x-a) 2 - ’ ' ( x - a f + i + -⅛- + Jk.
X b (x b) 2 ' · · ( x -bf ++ . ⅛.. + _L·l..
. + x-l + (x-l) 2 · · · + (as-Z) λ
The constants A 1 , A 2 . . . B 1 , B 2 . . . L 1 . . . L λ , can be determined by-ordinary algebraic methods, and each term is immediately integrable. The preceding is called the method of integration by decomposition into partial fractions.
The method here given applies also to the case where φ(x} = 0 has imaginary roots. In that case it is usually, however, simpler to employ a somewhat different treatment. Thus, to a pair of imaginary roots α±zβ corresponds a partial fraction of the form
Las + M
(as-α) 2 + j8 2 '
Also, for n pairs of equal imaginary roots, we have additional terms of the form
L 2 as + Μ., L. t as + M 3 L n x + M„
{(x-α) 2 +j8 2 ∣ a ’ {(z-a) a + 0 2 } 3 ” ’ * {(as-α) 2 + j 8 2 }" *
Each of these expressions consists of two parts, one of which can be immediately integrated. For example,
(L n as+M n )das L n (as - a)dx (L n a + M)dx {(x-af + β 2 } n {(x-α) 2 + β 2 } n {(≈-a) 2 + j8 2 }' l ’
the former can be at once found; the consideration of the latter class of expressions is postponed for the present.
Many integrals of the form here considered may he determined by a transformation, without the employment of the method of partial fractions.
For example, ∕~ -<L⅜ a L_
fax 2n + 2bx n +c is at once reduced to an elementary form by making x n ≈z. . Γ dx
Again ∕ —
J (x-a) m (x-b) n is reduced to depend on
∕*(1 -z) m+n ~ 2 dz J z m by making.
x — b Ex. 1. To find iff·r assume z =as 3 ;
,∕ as s -l then x b dx≈ ⅛zdz, and we get β y <Z∕ — A <q — A o Ex. 2. To find ∕* ⅛- .
J x(a + bx n )
Let x"~J, then ⅛=-l -⅛-, z x n z √ Γ dx 1 p dz 1 1 ( x n ∖ x(a + bx n ) njaz + b na ∖ a + bx n j 9 f dx J&(a+to?)‘
Let as 3 = —, and we get √ C dx 1 Γzdz z . b 1 , l ,, χ ∖ a + to*)” " ∖ ∕az+lΓ 3α + 3α 2 1 o ( &)
3αa∙ 3 3α 2 b ∖ af i )
1 - 2as 2 cos θ+x i
Here √ ^ dx f dx
1 -2as 2 cos θ+x i ~j∕(l + 2x cos ⅛0 + as 2 )(l-2as cos ⅛fl+as 2 )
1 ∕ ’ (as + 2 cos ⅜e)⅛c _ 1 ∕*(as-2 cos ⅜0)<Za
^ 4 cos ⅜0√ 1 + 2» cos ⅛0 + x 2 4 cos ⅜⅛∕ 1 - 2as cos ⅛0 + U
1 l + 2x cos ⅜0 + as 2 - 8 cos ⅛0 ° g 1 - 2as cos ⅛0 + x 2
1 i ,as + cos⅜0 , 1 as-cos⅜0) + 4Λp i ton ^⅛Γ +ta ° ^≡F!
1 , 1 +2x cos⅞0 + as 2 1 tan - 1 2⅛ , sιn ⅛θ
"~8 cos ⅜0 ° g 1 - 2x cos⅛0 + as a + 4 sin ⅜0 an l-as 2 Ex. 5. Find the integral of dx a + 2bx 2 + cas 4 when ac>b 2 .
It is easily seen from the last that its value may be written
1 √α+2⅜as + a! 2 ∖ ∕^ 1, 1 2¾√⅞ + κ 2
8∕c√α ° g √α-2κas + as 2 √c + 4√α(δ + κ 2 ) * ΙΠ √a-as 2 √c where κ= j ∖ ∕ x —- .
118. Several general classes of integrals can be easily reduced by a transformation to depend on those of the preceding article.
For example, the integral ff^dx reduces to^∕(z)-⅛ , by making c ax =z ; and, accordingly, it can be integrated whenever ∕(z) is a rational algebraic function of z.
Again, if we take tan |«=z, we get
2z l-z 2 , 2dz 8in x ~ ÏT? ’ cos a '~T+z 2 ’ dx ~ l+7 a and, the expression
/(sin x, cos x)dx, transforms into f(—— - ~ g2 ) j⅛L .
j ∖l+z a, l + z 2 ∕ l + z 2
Consequently, whenever /(sin x, cos «) is a rational function, the integration of /(sin x, cos x~}dx is reducible by the method of partial fractions.
119. Functions of this latter class are, however, usually more readily integrated by other processes. Thus, when /(sin x, cos «) is a rational and integer function, its integration depends on that of the sum of a number of expressions of the form sin” 1 « cos"« dx. As a number of other forms are readily reducible to this type, it is proposed to devote a short space to its discussion here.
In the first place it should be observed that whenever m or n is an odd integer, the expression sin”*« cos"« dx can be immediately integrated. For, if we suppose τι=2r + l, the integral transforms into
∕z m (l - z i ) r dz , by making z=sin«. Hence, as r is by hypothesis a positive integer, (1 -z 2 ) r can be expanded in a finite number of terms, and, the integral thus immediately obtained.
Again, if m + n be an even negative integer, the expression can be readily integrated; for, by assuming z ≈ tan «, we get r r [ λ 2t⅛-ι
/sin”·« cos"«d«= f~ n ∖ 1 + z 2 ) 2 dz .
This integral can be readily obtained by expansion.
120. When neither of these methods is applicable it is usual to find the integral of sin”*« cos n xdx by the method of successive reduction.
The formulae of reduction can be easily obtained by the method of integration by parts; thus J"sΛΑ m x cos n xdx≈∕~ cos"- 1 «sin”*««?(sin «)=/* cos ^ * aι c?(sin”· +1 «) cos" ^ 1 x sin m + 1 ∞ n - 1 ∕*. m , n „ „ , = 4 ∕ sιn m +⅛ cos" ^ 2 xdx.
τn + l 7>1 + I √
Again, f sin”*+ 2 « cos"~⅛‰ = f sin”*«(l - cos s x) cos n ^ 2 xdx =f sin”*« cos"" 2 «i?« - f sin”*« cos"««?« .
Substituting in the former equation, and transposing the latter integral to the other side of the equation, we get
∕* sin”·« cos"««?« = s * π + x cos —— + n -—1 /sin”·« cos"^¾α⅛ε, ∕ m + n m + nJ
Hence, when n is positive, the integral of sin”*« cos"««?« depends on that of sin”·« cos ’ *-¾fe. The corresponding formulae in which the degree of sin « is reduced can be immediately found.
It should be noted that these formulae of reduction are perfectly general, and hold whether m and n be positive or negative, integer or fractional. Accordingly, changing the sign of m, our first equation may be λvritten thus :—
/cos"« , cos" -1 « n — 1 Γ cos" -2 « , . m dx— —-, -- ∕ -, ÷- dx . sin”*« {m- 1) sin”*- 1 « m-lj sιn"*^⅛
121. These formulae of reduction, as well as many others, can be readily established by differentiation. For example, since
(sin”*« cos"«) = τn sin”* -1 « cos"+⅛-τι sin"∙+⅛ cos” -1 « dx = m sin”* -1 « cos" -1 « - (rn + n) sin"∙+⅛ cos" -1 «, the integration of the expression sin w *^ t ⅛ cos"-’««?« depends on that of sin”*" 1 « cos"- 1 ««?«; and similarly in other cases.
It may be noted that the integral (§118)
√ r ^ dx
{(« —α) 2 +3 2 } n is at once reduced to the class here considered by making« - a = βtan θ, when it becomes 1 ∕"ms 2 "- 2 0iZ0 .
3 2 "- 1 √
To find f tan"««?« .
Here f ta,n n xdx^f tan"^⅛(sec¾-1)«?« „ter^./tan"- 2 ««?« n- 1 J tan"- 1 « tan"- 3 « .
≈∙ —+&c.
71 - 1 71-3
Next, let us consider the integral ∕ d,x (a cos« + 6sin«)» ‘
Let tan α=≈4h , and we get a cos x+b sin «=(α 2 + δ 2 )1 sin (« + α).
Hence, making « + α—z, the integral transforms into
(α a +δ 8 )^ 2 ∕* -⅛-.
' J S111"Z
122. In many applications the results depend on integrals of the form here discussed when taken between the limits 0 and ζ . Such definite integrals are easily found when the indices m and n are positive integers.
7Γ
Commencing with the simple case of /2 sin"««?«, we have, since sin « cos"^ 1 x vanishes for both limits,
∕ 2 cos"«i?« = ?l—— ∕ 2 cos"- 2 «i?«.
Jo n Jο
By successive applications of this formula the definite integral in question can be always found when n is a positive integer; its form, however, depends on whether the index n is even or odd.
(1) Suppose n even, and equal to 2τ∙, τr o 1 7r then ∕* 2 cos 2r «d«=—— Γ 2 cos 2r ^¾fe, Jο 2t∙ ¾ and, accordingly, by successive applications, we get
∕∙-w- 2r , 1.3.5... (2τ∙-l) τr
∕ 2 c ° sjafe -2.4.β... 2r Τ'
(2) If π be odd, and equal to 2ι ’ + l, we get in like manner
∕*÷ cos 2f +⅛θ ’ *'j"',/ χ .
Jq 3.5.7... (2τ∙-f-l)
It is evident that in all cases
7Γ τr
∕ 2 sin"««?« = ∕ 2 cos"««?« ·
Jο Jo
(3) In like manner, we have π - π
∕∙ Λ> Ί ∕∙
∕ 2 sin’"« cos”««?« = ∕ 2 sin”»« cos"- 2 ««?« .
Jο m + nj ο
As in the former case, the value of this definite integral depends on whether the indices are odd or even.
First suppose n odd, and equal 2r +1, then ∕ 2 sin”*« cos 2r + 1 «c?« ≡— l o - = ∕ 2 sin”·« cos 2r - ~ i xdx. Jθ m + 2r + lj ο IT
Hence f 2 sin”»« cos 2r +W≈
Jο
2τ∙(2r-2). . . 2 . n .
= 7s r v τς i τ 7 Γβ∖ ∕ 1 Sill”'« COS « «?«
(2r + 77i + l)(2r+77i- 1) . . . (m + 3)∕ t
2 . 4 . 6 . . . 27-,
'"(τn + l)(τn + 3). . . (τn + 2r + l) * Next let n be even, and equal to 2r, then Γ— 2τ∙-l ∕-÷
∕ 2 sin 2 ”·« cos 2r ««?« — o . . ∙√ ∕ sin 2 ”·« cos 2r ^¾f≈ .
Jο 2(m + r)Jο
Hence, as before, f ^~sin 2 "⅛ cos2»·«i?« = - 1 --- ‘ ∕* 2 sin 2 ”*«i?«
¾ sιn x cos-r«««« - (2?ft + 2). . . (2m + 2r)Jο 1.3.5... (2r-l) .1.3.5... (2τ⅞-l) π “2.4.6 (2m + 2r) ' 2
When m and n are both fractional these definite integrals are reducible to Eulerian integrals,—functions of which a short discussion will be subsequently given.
The following examples are given for the purpose of illustrating the preceding results.
Ex. if (1 - x 2 ) m dix , where m is an integer.
Jo Am 2 ~ 4 ∙ 6 ∙∙∙
3.5.7... (2w + l)*
/--£ 9 , 3 . 6 . 12
Ex. 2. J λ cos 5 z , sιn⅛∞ . Ans. g—jf~i7'
π o ∕- 0 ° dx , Ex. 3. ∕ , 2 l 2 . w > where n is an integer.
√0 ∖ a +x I Ans * 1 ∙ 3 ∙ δ ∙ · · < 2 ^~ 3 )
2α 2n ~ 1 '2.4.6... (2n-2) *
7Γ
Ex. 4. ∕ 2 cos n + 2, ⅛ cos nx dx . Jo , g τr (n + 2r) (n+2r-l) . . . (n + r + l)
• 2 n + 2 ∙ , +ι 1.2.3 ... r Ex. 5. To deduce Wallis’s value for π by aid of the definite integrals considered in this article.
When m is positive, we have, for all values of x between 0 and—, sin 2m ^⅛> sin 2n ⅛> sin 2irt +⅛j accordingly, π τr π Γ 2 sin 2m ~ 1 xdx> f 2 wB m xdx> f~ 2 sin 2m ÷⅛z, Jo Jo Jo
. 1 . 3 5 . . . (2wt- 1) j r lies between 2. 4 6...(2rn-2)
2.4.6... 2m 2 3.5.7... (2m-l)
„ . 2.4.6... 2m d 3.5.7... (2m + l) ’
but when m is indefinitely increased the latter fractions tend to equality, and, consequently, we have the well-known formula of Wallis, viz. :—
π r .. „2.2 4.4 6.6
2 liπut ° f 1.3' 3.5' 5.7’’’
123. As a further example of the method of successive reduction, we shall consider the integral J^ r n, - 1 (l -x) n ~ x djχ.
Here, integrating by parts, we have
∕*T m - 1 (1 - x) n ~ λ diX = + ZL∑J t f c rn( i a .)n-2 ito .
J m m J
Again, J~ r m (l - x) n ~ 2 dx=J^ r n *^ 1 (l - x) n ~ 2 dx -J'x! m ~ ∖ l -x} n ~ l dx . Substituting, and transposing, we get f ¾ w ~ 1 (l -g) n ~ 1 d⅛= a^"(l -as) — ( — n —1 Γ -χ)n~ 2 dx . J m + n-1 m+n-ΥJ
By successive applications of this relation the proposed integral can be found whenever n is a positive integer. It can be determined in like manner when m is a positive integer. The integral of iC" t (α + lni) n dx readily admits of similar treatment.
The preceding is a simple case of the integration of what are styled i binomial differentials, i.e., differentials of the form x m (a + bx! n )ι dx, —in which m, n, p, q represent any numbers, positive or negative. We propose to determine in what cases such differentials can be immediately integrated by a transformation.
z ∖-L
Assume α + δiB n =zf, then x=( —Hi ∖ n , and we get P. n ρ+q-i to -"+ 1 ic n, (α + fo n ) s da!=—=Cz (zs-α) n dz. rib~ i ∙~
The latter can be immedately integrated when —— is an integer.
1 p
Again, substituting —- for x, the expression iC m (α + bx n ) q dx be- z - m -n ρ -2 — comes - z q (az n + b) q dz . This can be integrated whenever ——+ ^∙ is an integer.
n q
It can be shown that when neither of these conditions is fulfilled the integral of the binomial differential cannot be expressed except by infinite series.
124. Irrational Functions.— We shall next briefly consider the method of proceeding in the case of irrational expressions.
Suppose E(x,*fΕ)dx to represent the expression whose integration is sought, where F is a rational algebraic function, and X is a rational integer polynomial of any degree in ⅛. Here, since even powers of VX are rational, and odd powers contain VX as a factor, it is plain that F(ae, VX) can bo always reduced to the form
P + Q√X P + Q'√X ’ where P, Q, P', Q ’ are rational algebraic functions of x. Again, if this be multiplied by P'- Q'VX, it is reducible to the form
M + N√lζ ortoM+-¾,
√X
where M and N are rational functions. Consequently integrals of the proposed form are reducible in general to two parts, of which one is rational, and the other is of the form
//(a:) dx Φ&) √T'
It can be shown that, when X contains powers of x beyond the second, such integrals cannot be reduced to any of the elementary forms given at the commencement; and, accordingly, they depend on higher transcendental functions. When X is a cubic or a biquadratic, such integrals are reducible to elliptic functions, of which a short account shall be given below. When X is a polynomial of higher degree, the integrals are usually styled hyper-elliptic integrals. They were first treated of in a general manner by Abel.
125. We shall at present consider only the case where Xis a quadratic, of the form a + 2bx+cx i . The integral
∕ ∕(g) dx ≠( x ) ∖ la+2bx+caP can be rendered rational in different ways.
(1) First, let the roots of a + 2bx + cx 2 ≈= 0 be real, and suppose a + 2bx+cx! i '=c(x-a) (x-β).
If c be positive, we assume »—a = (ae —3)z 2 , or x ; then
√X-(-β)√⅛⅛, . dx 2 dz
' √χ β √T I-« 2 ’
Hence the transformed expression is a rational function of z. a I Rz⅛
If c be negative, we make a⅛ - ~-, and the transformed expression is rational, in like manner.
When the roots α and β are imaginary this method of transformation introduces imaginary forms into our results. In such cases it is usually more advantageous to adopt a different treatment.
For instance, if we assume
√α + 2bx + cx q = z - x Vc^, we get a + 25ae=z 2 -2a»Vc.
it z 2 -«
Hence ≈=- 7 τ=∙ »
2(δ+z√^)
, dix dz and ; ≈ -
∖ Ja + 2bx + cx i 5 + zVc .
This substitution consequently furnishes a rational function in z.
Again, when c is negative the expression becomes rational by the assumption ∖ ∕a + 2bx + cx 2 ∙~ ∖ fa + xz.
In general, if we substitute ⅛ ^ or x > where λ, λ', μ, fe ’ λ'+2√z + √z 2 μ ’ , V, ν' satisfy the equations ∕z 2 - Kv = a, λ!v + Kv - 2μμ' = 2b» μ' 2 - χ' l ∕' ∙= c, it can be shown without difficulty that dx 2dz a + 2bx + cx i λ' + 2μ'z + √z 2 aud accordingly the function f(x) dx Φ<f) ^fa + 2bx+c^ i becomes rational by this transformation.
This last is a particular case of the general method adopted by Jacobi (Fundamenta nova theorise f unctionum ellipticarum) for the transformation of elliptic integrals.
126. The class of integrals here discussed admits also of another mode of treatment.
Thus it can be shown that, if F(x) is an integer rational function of the degree n, then Γ.. ⅞)l⅞i = n Γ H h - + ≠(ic)√α+2ta+cx 2 , J √a + 2fec + cx 2 J Va + 26x + ca∙ 2 in which a is a constant, and ≠(α) is at most of the degree n -1 in X. For, if we differentiate the expression x m ∖ ∕a + 2bx + <x~ with respect to x, we readily obtain, after the integration of both sides, and the substitution of X for a + 2hx + cx 2 , Γxi m +idx . λ ... pX! n dx ι ∕~x m ~ 1 dx a^√X = (τn + l)c∕ + ( 2m + 1 >y √χ + mc J "ifχ ' ’
Hence, making m≈0, 1, 2, 3 . . . in succession, it is easily seen that ∕⅛⅛ is expressible in terms of ∕⅛= and of an algebraic j √x √√χ
expression of the form φ(∙r)VX, where φ(a∙) is of the degree n -1 at highest.
Again, by the method of partial fractions the integral r∕(¾) dx J Φ(, x ) ∖ ∕a + 2bx+cx i reduces to terms such as the preceding, along with terms of the form √ l ^ dx (x - α)' l √ a + 2bx + ∞ 2
If in this latter we substitute — for x - a, it reduces to the form z √ ^ z n ~ 1 dz
√A+2Bz + Cz 2 ’
in which A = c, B=-δ-cα, C = a+2δa + ca 2 .
127. Integrals of the form here discussed may also be treated by the method of indeterminate coefficients. Thus, writing X for α + 2bx + ∞ 2 , and differentiating the equation at the commencement of §126, we get
F(ae) α λ. j .ι ∖ κΓV , ≠(a 0(δ + ∞) √r"√χ- +φwvx+ ~√χ—
or F(x) = a +≠'(aj)(a + 2bx + ex 2 ) + ≠(x)(δ + ex).
Hence, by equating coefficients of like powers of x, the value of a and of the coefficients in φ(x) can be determined.
For example, let it be proposed to find √ - x 2 dx ∖ ∕ a + 2bx + cx i
Writing ∖ + 2μx+vx 2 for φ(x), we get a? = a + 2(α + 2bx + ex i )(μ + vx) + (λ + 2μX + vx 2 )(b + ex), from which we deduce
1 5b . 5δ 2 2a b κ τ 1 2∖
r= 3?’ M= "12?’ λ "2^^"3^ , β ≈2^< 3αc " 5δ )∙
128. Again, if F denote a rational function, the integral t ∕F(z, ∖ ∕ax+b, *fa'x + V)dx is reducible to the preceding type, by making Vαα + 5=∙ι∕. For this gives
<fc- 2 ⅛, √5⅛+⅛- ∕⅛e + ≤L∑≤ i ;
α V a a and the proposed becomes of the form
∕∕(y, √Y)⅜, in which Y is of the second degree in y.
129. Having given a sketch of the various methods of reduction of integrals to the forms usually regarded as elementary, we proceed to introduce further transcendental integrals by considering the integral f c∞⅛ildx , in which/(a;) and φ(x) are rational algebraic √ φ(x) functions of x.
By the method of partial fractions we may write
φ^ = F(≈) + 2— + · · · + s (a5 . α )n + ι ’
or, making a slight change in the constants,
= F(x) + SA(z - a) - 1 + 2 A 1 ⅛ - a) * 1 +... + 2 A J ½- ∖∖ x - α) - 1 φ(x) dx ∖ dxj = F(x) + 2(A + A 1 D + A 2 D 2 + . . . +A n D∙)(as-a)^ 1 , where D stands for the svmbol .
dx
The method of integrating ^F(x)e n3c dx has been already considered (§ 116). The integral of the remainder depends on that of the expression e n ⅛c(Λ +A 1 D + A 2 D 2 . . . + A n D n )(aj-α)- 1 .
If the symbolic expression A + A 1 D + A 2 D 2 . . . + A n D" be represented by ∕(D), this integral, in symbolic notation, is represented by
D- 1 e"7ID)(aj-α)- 1 , or, by § 89,
D-y(D-n)Jll .
x— a
Again if f( - n), or A - A 1 π+A 2 ra 2 - . . . ± A w n n , be represented by N, we have
ΛD-n)-N-^D+gD>- ... +A.D-.
Hence, observing that N, , . . · are indeι>endent of x, dn dn 2 we have l>-⅛y(D)-L - D->! N-gθ + . . . + A.Di = y ∕ e nx dx <fN e ,u ≈ d 2 N d ∕ e nx ∖ J x-a dn x-a dn 2 dx ∖ x-a)
÷<∏^)∙
Consequently, the class of integrals here considered depends ultimately on the integral ∕ e n3c dx x-a'
If we make x-α=log z, this integral reduces to the form (§ 115, Ex. 4)
f dz
√logz '
It is impossible to represent this latter integral, in a finite form, in terms of z. It is accordingly regarded as a function sui generis, and is usually styled the logarithmic integral, and sometimes Soldner’s integral. Its expression in the form of a series will be deduced in a subsequent section.
130. Next, if we replace n by in, where i stands for V-Î, e n * ff) becomes (cos nx+i sin nx) , φ(z) φ(x) and by an analogous treatment it can be proved that integrals of the forms
/cos nx ∙-4⅛- dx and Γ sin nx dx φ(x) J Φ( χ ) depend on the forms
/cos z dz an ^ ∕ *sin z dz z J z
Finally, denoting by F(sin x, cos x) an integer polynomial in sin x and cos x. it can be shown that the integral ∕ ∙⅞fX F(sin x, cos x)dx φ( x ) can be reduced to the same fundamental forms. For tho polynomial F(sin x, cos x) can be transformed into a linear function of sines and cosines of multiples of x. Again, decomposing Φ( x ) by the method of partial fractions, the integral in question can be made to depend on integrals of the form
/sin mx dx . f cos mx dx
(x-α)"+ 1 J (z-α)-+ 1 '
and consequently on ∕ dx sin mx (~ ∖ n -—— r and ∕* dx cos mx () - --- - . ∖ dxJ (x-a) J ∖ dx) (x-a)
These integrals, by the method of §116, depend on f . C⅛)" sin,ω ∕' (⅛)" cm ° ’ ≈
√ x-a J x-a and, consequently, on the forms
/sin z dz , ∕*cos z dz and ∕
z J z
131. These latter integrals also are now regarded as primary functions in analysis, and are incapable of representation in terms of z except by infinite series. , .
These functions have been largely treated of by mathematicians, more especially by Schlomilch (Crelle, vol. xxxiii. ), by whom they were styled the sine-integral and the cosine-integral. Also, introducing a slight modification, the logarithmic integral can be written in the form ∕ e~*dz z In this latter shape it is called the exponential integral.
Hence, adopting Schlomilch’s notation, we write si x≈r x si - n i ι ^ du, Jo z Jο u
Ci x-f x ^-dz-f 1 ⅜ d u, J* ≈ √∞ «
_ Z" ~ x fi~* ∕* ^ 1 Λ-WX
Eiae=/ — dz=∕ — du. J< β * J∞ U
Again, if Liz=∕* r ⅛-,we have Jο log«
Lie≈=Eia3,
An interesting and valuable historical account of these transcendental functions is given by Mr J. W. L. Glaisher in the Transactions of the Royal Society, 1870, of which want of space prevents our giving a fuller account. Mr Glaisher has also, in the same memoir, given tables of the numerical values of these transcendental functions for a number of different arguments.
It may be added that the logarithmic integral was discussed, and tabulated by Soldner in 1809.
Numerous integrals have been reduced to depend on the foregoing transcendents. For example, in the great tables of Bierens de Haan {Nouvelles tables d'intégrales définies, Leyden, 1867) nearly 450 forms are shown to be reducible to one or other of the functions considered in this section.
What has been said here will help to exhibit the way in which the necessity for the introduction of new transcendental functions arises as the calculus is developed, and to show that around each new transcendent whole classes of integrals are grouped.
132. The very limited number of differentials which can be integrated in a finite form by aid of the ordinary functions makes it an interesting and important question to find whether the integral of any proposed differential expression is capable of being represented by such functions or not. This problem appears to have been first discussed in a general manner by Abel. Our limits admit only of a statement of one or two of the general results thus arrived at. The reader will find a tolerably full account of the treatment of the question in Bertrand’s Calcul Intégral, pp. 89-110.
Abel’s fundamental theorem may be stated as follows. Suppose y to be an algebraic function of the variable x, that is, a function defined by a rational equation F(ae, y) = 0, which is of the nth degree in y ; then, if the integral fydx be also an algebraic function of X, it must be of the form fydx = P o + P 1 y +P 2 y 8 + &c. + P n _ ιy n - 1 , in which P o , P 1 , P, 2 . . . P n ~ι are rational functions of x.
The functions P o , P 1 . . . . can be investigated by the method of indeterminate coefficients, which, in the great majority of cases, will show the impossibility of an algebraic integral.
In the particular case where y— χ∕X, X denoting a rational function of x, it has been shown by Liouville, as a consequence of Abel’s theorem, that, if the integral fdx ’ ^∕X be algebraic, it must be of the form P 1 "ζ∕X, in which P 1 is a rational algebraic function of x.
Again, denoting X by and substituting T for M n ^ 1 N, if the integral
/Μώ
where M and T are whole polynomials, be expressible algebraically, Θ
it is of the form -,-. r ~ , where Θ is another polynomial.
√ P
If the equation
/Μώ Θ
χ∕τ~
be differentiated, we see that the highest degree of x in Θ must be one greater than that in Μ. Hence, by the method of indetermihate coefficients the integral, if it is algebraic, can be found; or else it can be shown to be impossible under such a form.
Again, if t, u, v, . . . be algebraic functions of x, the differential of t + A log w + B log v+kc., where A, B, C are constants, is evidently algebraic. The converse theorem was investigated by Abel, viz., when y is algebraic, to find when fydx can be expressed by algebraic and logarithmic functions. He showed that if fydx=t + k log u + B log ι> + C log w + &c., then the functions t, u, v, . . . are capable of being expressed as integer functions of y.
Abel’s theorem was extended by Liouville, who started from supposing fydx=Y{x, e n , e ν , log w, . . . ), where u, v, w, &c., are algebraic functions of x. He proved that, when y is algebraic, the expression for its integral cannot contain an exponential, such as e ,, . Also that a logarithmic function, such as log w, cannot enter into the integral except in a linear form with a constant coefficient.
' ∕~^Pdx
In particular, it is shown by Abel that whenever J fjff is expressible explicitly, it must be of the form
∕^≡ ® +λ log i±⅞^S+B log r⅛∕∣ +ji J √R √R α-0√R 7-δ√R
in which P and R are integral polynomial functions of x. Definite Integrals.
133. The investigations have thus far been chiefly limited to what are styled indefinite integrals. It is plain from §107 that, whenever the expression <p{x) remains finite between the limits of integration, its definite integral, taken between those limits, can be determined whenever its indefinite integral is known.
For instance, since ∕ dx 1 , 1 Λr+cosa∖
-— «-=— tan* 1 I —7 ) ,
1 + 2x cos a + x i sin a ∖ sin α ∕
we have Γ 1 dx 1C , 1 + cos a , ι cos α) a
∕ =—x s = -s ∖ tan ^ 1 —·. tan ^ 1 -,— >=— : — .
√0 1+2z cos <x + x 2 sin a( sin a sin α) 2 sin a
Also (Ex. 8, § 113),
/ifø 2 , 1 ∕, a, ae ∖
≈ _— tan^ 1 tan — tan — I .
1 + cos a cos x sin a ∖ 2 2 ∕
Accordingly ∕* — = ——; Jo 1 + cos a cos x sin a f π dx ιr 0r Jo iT*cosaΓ √Γ∑> ’ ' vhen k< 1 ∙
From this we readily get π f I dx Jo α 2 cos*⅛ + δ 2 sin¾ = 2αά ’
134. As definite integrals have frequently to be considered in which we regard one or both of the limits as infinite, it is necessary to determine whether the equation
∕-x
F(≠≡F(X)-F(¾) ∙∕∙ 4 o holds for infinite limits.
Suppose when X becomes infinitely great that F(a∙) approaches a finite limit, represented by F(α>), then
χ 1 T a , jF(X)-F(z 0 )l -F(a>)-F(a ⅛).
Consequently the formula holds in this case.
In like manner if, when x becomes - oo, F(ae tends to a finite value F( - oo ), we have
/* X F'(ae)dae=F(X) - F( - oo ).
∙× — co
∕∙÷∞
Also ∕ F'(z)<fø=F(oo ) - F( - ∞ ).
J — co
Hence, when F'(ie) remains finite between the limits, and F(ae) has determinate values for both limits, the equation
∕‰^=F(z)-F(a⅛)
always holds.
For example, in the integral
/cfø 1 . ( x ∖ a 2 +x 2 ~ a an ∖ a ) ’ when x≈∞, tan- 1 (~ ∣ has for its limit - 7r and when x — -∞, ∖ aj 2
tan^ 1 — has for limit — — · hence a 2
∕* χ, dx _ π ∕* x dx _ η Jo a l +ti i 2α ’ J a l +x 2 ~ a
—oo
Also, from the integrals given in § 114, we get
∕∙aθ n B x b ∕ c~ ax cos bx dx≈ ——— . ∕ c~ ax sin bx dx≈ ——- . Jo a i + h i Jo a i + b 2 Again, in Ex. 4, § 113, we have
F(0)= 7 L--, F(α>) ≥⅛-, (ac - b) ∖ ∕a ac ~ l> i Γ a dx = 1
Jo (a + 2bx + cx 2 )^ aci + bai
In like manner, from Ex. 5, § 117, we get
∕" = — , where h = 2( v∕αc + b). y 0 a + 2bx i + cx i 2^f a h
It may be noted that if F(x) approaches a finite value F(∞ ) as x approaohes α> the derived function F'(a:) must vanish at the same time.
135. As a further example, let us consider the definite integral j ≠(ir)X n d<r, where <∕>(z) is an arbitrary polynomial of the degree n-1 in x, and X n is the coefficient of α" in the expansion of (1 - 2ax + a 2 ) ∣ .
It has been shown (§ 56) that x -2~,∙ 6 1 ,..2. (⅛)>- 1 >∙∙
Again, by the method of § 116, we have y dxφ(x) (^ n (x 2 -l) n =Θ + (-l)βdx(x 2 -l) n moreover, ( J- ∖ ≠(x) = 0 by hypothesis, and when the limits ∖CKB∕
+1 and -1 are substituted each term in Θ vanishes separately; hence we have
Z+ι
φ(x')X n dx=0 .
From this it is readily seen that so long as m and n are unequal we have
Z·+l , XmXndX — 0 .
1
136. There are many integrals which are capable of being determined between certain definite limits without any previous knowledge of the corresponding indefinite integral, and even in cases where the consideration of the indefinite integral would lead to the introduction of a higher transcendental function. Examples of this class will be met with further on.
137. Next, reverting to our original definition (§ 106), we have
√1x)⅛=lim.^ 1 -¾)∕(z 0 ) + (¾-≈ 1 )∕(¾)∙∙ +(X-a⅛-ι)∕(a⅞,-ι)], in which f{x) is supposed to be continuous between the limits x 0 and X. If now A represents the least and B the greatest value of ∕(x) between these limits, it is plain that
(¾-⅞)∕l⅞) + (⅞-<⅛) + · · · + (X-a⅛-ι)∕(a⅞,-ι) is greater than (X - a⅛)A, and less than (X - α⅛)B.
∕∙χ
Hence ∕ fo e ) dx =(χ _ x ) M , -∕∙ r o where M > A and < B .
Again, when f{x) is a continuous function, in passing from one limit to the other it varies so as always to lie between the values A and B. Consequently for some value ξ, of x, we must have∕(ξ) = M, where ξ lies between a⅛ and X, i.e., ξ is of the form x 0 + θ(X-xj), where θ is positive and less than unity. Hence, whenever f(x) is finite and continuous between the limits x 0 and X, we have
^‰<X- a ¾)∕{v*(X-* 0 )} -
In like manner it is shown that j χ9 ∕(*W)<‰V{⅜ + 0(X-≈o)j∙ φ(f}dx, provided j ∖ x) and φ(x} are finite and continuous between the limits x q and X, and φ(x) has always the same sign between these limits.
For example, let φ{x)=- —, and write fx) instead of then X — CL X ~~ CL
∕ /(«)<&>=({-»)/({) log ^ χ ~~ l , in which we suppose that X - a, x a -a have the same sign, and ξ lies between x 0 and X.
In particular, if α=0, we have J x f(x)dx≈ξfξ) log — .
Jxa ⅜
138. Taylor's Theorem.— The method of definite integrals furnishes us with a simple demonstration of Taylor’s series.
For, if in the equation ΛX + A)-∕(X)=∕ x+ * f^dx we substitute X + h - z for x, we get
∕(X + Ä) -ΛX) =∕∕'(X + h - z)dz . √o
Integrating by parts, we have β ∖ X + h - z)dz = s∕'(X + h-z) βf ∖ X + hz)dz; hence j↑X + h) -ΛX)=Λ∕(X) + z ∕*∕"(X + h - z)zdz . Again, ∕^ a h 2 ∕^ a ~2,7 y
∕''(X + 7 i - s )afe≈√U-∕'(X) β f "' ( x + h . z ) z HL· i and so on.
Hence we get finally
∕[X+Λ)=∕(X) + ⅞∕'(X' + ⅛Γ(X) + . . .+ r ^∕<-n(X) J. J. . 4 ∖ !l∕ — 1
√∙' 1 - 1 (x÷ ,ι -->p⅛'t
Accordingly the remainder, R n , after n terms, in Taylor’s series, is represented by the definite integral
This value of R„ can be identified with that given in § 46, for by § 137 we have
TT ∕∙* 7>n
R n =^÷ i ∕ z n - 1 ⅛ = U i - , I W - 1 √0 ∖ nf where U lies between the greatest and least values of f n ∖ X + h- z) between the limits 0 and h for z.
Hence, since any value of z between 0 and h may be represented by (1 - θ)h, where θ > 0 and < 1, we have h n ^n=^-H n KX + θh). I2L
139. Thus far the function fix') under the sign of integration has been supposed to have a finite value for all values of x between the limits of integration.
Let the indefinite integral of f(x)dx be denoted by F(ae), and suppose fx) = ∞ when x≈.a, where a lies between the limits X and x 0 ; then, decomposing the integral into two parts, we have f*f(x)dx = Γ f{x}dx + h™' ∕^*f(f)dx ⅛Z xq *J xq Jx
- [⅛w-ιw] + [i ’ (X)-⅛(*>] .
Accordingly, whenever F(α) has a finite and determinate value, we have
∕‰fc=F(X)-F(α⅛).
This result also holds if ∕(ic) becomes infinite at one of the limits, provided F(ae) is finite and determinate at the same time.
For example, the expression —„ > becomes infinite r √(α-z)(z-0)
when x≈a, and also when but (Ex. 6, § 113) F(α) = π, F(3) = 0, ∕ a dx ^{ a -x){x-β')
140. The complete discussion of the exceptional cases in definite integrals is due to Cauchy. We purpose here to give a brief account of his method.
Suppose that the function f(x y ) becomes infinite for the particular values of x represented by x ll x 2 , . . . x n , lying between the limits of integration; then we have
∕" x ∕(α)ifo= ∕^ ∖ f(x}dx+ f i f(x)dx . . .+ f x f(x')dx fX( i Jxn J x ∖ . -' x n ( ∕"Xl~ Ml e ∕*¾ - ∕ i 2 e =lim. of ∕ f(x)dx+ ∕ ∕(αj)d⅛+ . .
JXo J χ χ+V∖t + ∕* x f{f)dx I , ∙∕ Xn+vne where β denotes an infinitely small quantity, and μ i , v i , μ 2 , v, 2 , . . . v n , are positive constants, but arbitrary.
In addition, if the limits X and x 0 become + ∞ and - ∞, we write ∕ +50 Λ*> = lim∙! Λ l " μ %∙X≈+ A 2 μ2 %)dx. . . √-αo ' J __i_ Jx v ∖ -v ∖ κ μ∙κ
¾.
+∕ ve f{x)dx ∖ , J Xn+vne ’ in which μ, v are new positive arbitrary constants.
In all cases, the general values of the definite integrals J*f{x)dx, ∕~ + flx)dx , Jx a J -00
deduced from the preceding equations, depend on the form of the function /(ae), and may be finite and determinate, or infinite, or indeterminate, depending on the values attributed to the arbitrary constants μ, v, μ 1 , v 1 , . . . μ n , v n .
Whenever the integrals become indeterminate, if each of the constants μ, v, . . . μ n , > , n, be made unity, the corresponding values of f*j↑x)dx and ∕^ + ftf)dx J ∙⅞ J —00
become r— ∕"Xl~ e ∕'X∙>-ι ∕-χ -η lim. ∕ J ∖ x}dx+ ∕ " J ∖ x~)dx . . . + Γ f(x)dx '-Jχ 9 ' 7 X i +e Jχn+e "* and li m ∙ Γ ∕* fx)dx + f 2 f{x)dx . . . + i* f(x)dx~l · L-∕- 1 Zr 1 +e Jχn+e J
These are called, by Cauchy, the principal values of the definite integrals ∕~ x fx)dx and ∕~ + ∞fx}dx , Jχ⅛ J -∞ in the case in question. Again, the definite integral f J↑x)dx, Ja if ∕(ic) be finite when aj=α, is infinitely small if the difference between the limits a and b is an evanescent quantity.
But, if f{x) become infinitely great at the same time, the value of the definite integral may be finite, or even infinite. In the latter cases the integral is called a singular definite integral.
For instance, if fx 1 ) = ∞, the integral
∕∙jr 1 -u>e ∕ fx)dx, Jx l -e where e is an infinitesimal, is of this class. Its value may be represented by the method of § 137; for, if f 1 denote the limit of (ae - x 1 )f(x 1 ) when x=x 1 , we have ^χ l -μ 1 ι
∕ Λ≈)^≈∕ι log Mi ·
∙⅞l-<
∕0! 1 +e i ff)dx=f x log— . .+∙ , ι β rι
Again, if the limits a and b each become infinite, while preserving the same sign, we have another class of singular definite integrals, such as Γ vt f{x~)dx , ∕ ι e f(x)dx , y ⅛ ∙ z -⅛ in which e is considered evanescent as before.
In this, as in the former case, if xfx) tend to a limiting value f, when X is infinitely great, we shall have 1 _ 1 v, f(f)dx=f log (y ) ,y r , '∕(x)c‰=√lθg μ . ^i ~μe
141. We shall illustrate the preceding by a few simple examples commencing with the definite integral ∕" x dx J —Tr. ae 1 .
Here the function — becomes infinite when ae=0, and we have X Z x dx Γ⅛ dx + ∣ ^^Xdx x a ^ x J-χ a ~ x Jo ~ x -lim. Γ ⅛ + J ⅛Ι , L√-⅛ ≈ J·’ ≈J
but
∕∙X dx . X Γ~V^ dx fy^ dz 1 ∕ue∖
∕ T" log 77 ’ ∕ ^“7 T*" log (≡^) i J ve x ve J-X a x JX t > z ∖ a3 0∕
- lta ∙[ZU⅞÷Γ¾W÷^(f)∙
Accordingly, the principal value of f — islogΛ-), and its J-x a x ∖ x ο∕
X
general value is log flog — · The latter is perfectly arbitrary a⅛ v and indeterminate.
Again, each of the singular definite integrals
1
∕* e dx f ve dx J e ^ ’ J'- is equal to log — .
κτ f^ s - dx 1 . Γ f~r^ dx f^X∙ dx~ ∖ fct f^- lιm LH. ^ + f d∙ d i /—Μ« dae 1 1 ∕ x dx 1 1
√- a ∙ 0 x i μe ⅜ Λe x t,e x ∕* χ dx ,. Γ 1 , 1 1 1 Ί
•’Zx. m, Lμe ye X a¾J ’
Consequently, the principal value and the general value of the definite integral are both infinite in this case.
In like manner ∕- χ dx r 1 ∕ 1 1,1 1∖
x∙ 3 ~ m, i k *' 2 e 3 M 2 ® 2 + ¾ 2 χ2 ) '
Accordingly, the general value of the integral is infinite, while its principal value is ⅜ .
Next let us consider the singular definite integral
∕7β (x - a)dx (x-a) 2 + b 2 " Here ∕⅛-^.y⅛~ i ⅛{<≈-) , + i2 } ·
If we substitute the proposed limits, and afterwards make e = 0, we readily find the value of the proposed to be log J—an indeterminate quantity, as v is by hypothesis supposed to be arbitrary. Likewise
∕ + ° 0 JΞZ⅛⅛.=lim. A ^ x ~^ dx ≈ log A , when e = 0. 7_ oe (x-a) 2 + b 2 -α) 3 + ⅛ a b v
Accordingly the general value of ∕-+ 0 ° (x-d)dx J-< ο (a≈-α) 2 + 0 2 is perfectly arbitrary, while its principal value is zero.
In like manner, since
∕ ⅛ = ltan-√^V
J (x-a) 2 + b 2 b ∖ b J , we find the general and also the principal value of ∕*+ c0 dx ∙π {x-af+b 2 ^b'
Again, it readily follows from the last result that, when ae > δ 2 , the value of the definite integral ∕" co dx ιr
√C 00 a + 2bx + cx 2 18 √ flc -js ’
142. Next let us consider the definite integral u=Γ ^~^ dx. Jο x
Here u ≈ lim. ∕ Φ(ax}~Φ(bx) dx . J vt x
1 a
But f μe Φ( ax ^ dx = Γ μe , making αx- z. -∕ ye X Jave z
Also f μe ΦW d χ ≈ Φ^ dz . y>ze X ''bvt z
. ∙. «=lim. f μi ≠ω⅛im. f bvt , √ 6 z J av < z
μ«
where e is infinitely small.
The limit of the latter integral, as already seen, is ≠(0) log J . Also, whenever φ(z) tends to a definite limiting value φ(∞ ) when z is infinite, we have μt φ(f)dz = ^ 00 ) log .
μe
.. ^-°≠⅛ g )-≠(to)⅛. w „) _ Λ0)} 10g (I), in this case.
Cb
Again, whenever· ∕^ μe Φf~)djj j a zer0j we have
*∕ δ *
pê
√0 x ∖ a J
In the latter form this result is called Frullani’s theorem, having been communicated by Frullani to Plana in 1821, and subsequently published in Mem. del. Soc. Ral. , 1828.
These results, though limited as to their generality, contain many particular integrals under them.
For example, since e~ ax becomes 0 when x=∞, and 1 when x=0, we have ∕∙∞ e -ax^ e -tz h ∕ ———— dx ∙= log
√o ae «
Again, when x = 0, tan~ 1 (αic) becomes 0; and when x≈∞, tan^ 1 α≈ = - . Consequently we have
2
∕° 0 tan^ 1 αa!-tan^‰ , π 1 a dx≈= — log -.~ ·
x 2° b
Also, from the periodic character of cos z, it is readily seen that *
∕-⅛ cosz dz J b Z v* vanishes when e = 0.
Hence ∕°° cwflg -.⅛⅛- log^ .
√o ® b
In like manner we have P° 1og>÷^- aa )-log(^+^- tae ) j r =ι 0g Λ + w∖ c l Jq X ∖ « ) b
Frullani’s theorem has attracted considerable attention recently, and many remarkable applications, both in single and multiple integrals, have been given by Mr Elliott, Mr Leudesdorf, and others, chiefly in the Proceedings of the London Mathematical Society 1876, 1877, 1878.
143. The consideration of singular definite integrals furnishes a method for the calculation of the general value of a definite integral when its principal value is known.
Thus, if A be the general value and B the principal value of ∕-X
∕ fx}dx, where/(«) is supposed to become infinite for the values *Z ∙Γq x 1 , ¾ . . , x n of x, then the difference A - B, from the preceding investigation, will consist of the sum of the singular definite integrals Z rl~μe ∕∙x l +e f(x)dx, ∕ ff)dx, . . .
—e -∕^ 1 +ι∕ 1 e
Consequently if ∕ 1 , ∕ s . . . f n , as before, denote the limiting values of
(≈-¾W)> (x-a⅛)∕(a!), · · · (aι-a⅞ 1 )∕(x), when x ==aq, x≈x i , . . . x≈x n , respectively, we shall have
A-B=∕ 1 log 'h- + ∕ s log *≡+ . . . +∕ n logi⅛.
rl v 2 v n
Accordingly, in order that the definite integral ∕ ftxflx should have a finite and determinate value it is necessary that the quantities ∕ 1 , ∕ 2 . . . f n should each be evanescent.
When the limits X and x 0 are + ∞ and - 00, to the value of A - B here given we must add the term f log H , provided xf(x ∖ as V x becomes infinitely great, tends to a definite limiting value f.
144. For example, if .∙Λ⅛ , be a rational algebraic function, then F(z)
the integral f 'Qr⅛lx has a finite and determinate value, pro- 0 √-α> F(αr) 1 vided (1) the equation F(ae) = 0 has no real roots, and (2) the degree of x in the denominator F(a?) exceeds that of the numerator fix') by two at least. For the former condition implies that $$ does not become infinite for any real finite value of x, and it follows from the latter condition that x J~) becomes evanescent when x F(x) becomes infinite.
In order to find the value of ∕* ~⅛x * n case > we su P'
J— ∞ "W
pose
A(aj-α) + B (z-α) 2 + ⅛ 2 to represent the partial fraction corresponding to a pair of conjugate roots α±⅛ of the equation F(ae) = 0; then, as we have shown that the general and the principal values of the definite integral are the same in this case, we may write
∕ +0 °‰-iim. ∕ + ÷⅛
√-oo F(*)∙ √. JL
€
1 ∙ - ∕' + ^ e ^A(re-α) +B
- 1,m ∙v i €
But we have already seen that ∕" + ^ (x-a)dx _ , ∕ + ~ da _ rr J_x_ (x-aY + h i , (a≈-α) s + δ 2 ~ b · e e
Consequently
∕∙+° 0 ∕(α), Z∞ W x &
=√Ιr+⅜+∙∙∙ ⅜0> ∖ bl bi bn J where B 1 , δ 1 , B 2 , δ 2 . . . B„, b n represent the constants corresponding to the n pairs of imaginary roots in the equation F(z) = 0.
As an example, let us consider the definite integral
∕*+∞X 2m (fe J~ v > 1 + ae 2 «’
where m and n are positive integers, of which n is the greater.
By aid of the theory of equations it can be shown without difficulty that B l , B 2 B re
⅛ι ’ b i ’ ’ - ' b n are respectively equal to sin θ sin 30 sin (2n-l)0 , , . . . , η η n
, λ 2rn + l where θ -= —-τ— π. 2n
Consequently we have
Jk + Jk . . . + ⅛ = -[sin 0+ sin 30+ . . . + sin(2n-l)0] b 1 h i bn '><'
1 ∖ sin ¾+l√ 2n
Z+ ∞ ∕p2rnz7 p σr f+^ c 2rn+l ·
∞ 1+a ^ ns i n ∙⅛±2ιr 2n
Hence it follows immediately that ∕ ° 0 x 2m dx π 1 +sc in ~ 0 ,, -2m+l 2n sin π
2n
By a corresponding investigation it can be shown that ∕* 00 ir 2m da! π j 2m + U l-ar*« = 2n cot ~2n~ v '
These results are readily transformed into f co x a ~ 1 dx π , f^x^-^ i dx . ∕ , and ∕ = π cot ατr, y 0 1+x sin aπ √ 0 1-x where a is less than unity.
A few simple examples are added.
∕-∞ dx 7r
(1) To show th∙t j ∕ ⅛ to .
(2) If n be less than unity, the equations f i x η + χ- η . π nιr ∕ 1 x n -x~ n , ιr , nιr can be readily established.
(3) If a < ιr, by a simple transformation it can be shown that y r∞ e a x ~ e - ax a I —- ,αx=⅜ sec- r , √ 0 e ιrx + e~ vx 2
f x c ax_ e -ax a ∕ dx≈⅛ tan — .
J o e πx -e~ πx 2 2
(4) If a + b < π, we can prove the equation ∕∙° 0 {e ax + e~ ax )(J x +e~ bx ) = 2cos 2^ c08 y JQ e ,rx + e~ ιrx cosα+cos⅛
(5) To find the value of ∕^ 1 dx Jo (l-® 6 )*’
Assume 1 - α β =—ι- a , l + 2∕ β log (l-ic β )= -log(1 + 3∕ β ), x?dx ifdg V X is ≈ a 1 - 5 ,-tA i but,»-— t .
∕* 1 dx f^ tκ dι∕ _ τr Jo (l-a β )*~Λ i + r = 3 *
(6) In like manner J —— i -= — . y ° (l-α 2 ") 2 ~* 2nsin ⅛
145. We now propose to consider some of the general methods of evaluating definite integrals. It is obvious that the value of the definite integral
∕∙δ
j a fJ>fx is independent of the variable x, and is a function of the limits a and b, as also of any constant parameters contained in the function f{x). We proceed to show that a definite integral may be differentiated, and also integrated, with respect to any such parameter.
Differentiation under the Sign of Integration.
146. Suppose the function/(ae) to contain a constant parameter a; i.e., let /(«) = </>(«, a); then, denoting the definite integral by u, we have u = ∕ φ(χ, a)dx. ja
Also, let the limits a and b be independent of a ; then, if Au denote the change in u arising from the change ∆α in a, we get ∕* δ r ι
δk = Z ( ≠(a ¾ a + δ ≡) ^ Φ( x > «) ∫ die,
. Δ¾ f^l> φ{x, a + ∆a)- φlx, a} ∆a Ja Δα
Hence, passing to the limit, dff=Γ b dφ(x, ct) t fo, da fa da
This principle is called differentiation under the sign of integration, and, by aid of it, from any known integral a number of others can in general be determined by differentiation with respect to the constants contained in the integral.
For example, if we differentiate the equation ∕*° 0 1 ∕ e~ ax dx=— , Oo a we get Γ c - ax xdx= ∖ ; Jo a i and, by n successive differentiations, /*“β , „ , 1 · 2 . 3 . . . . n Jo a n + 1
Again, if the equation ∕→ cc dx π J_ χ a+ 2bx + cx 2 ~ √ ac be differentiated with respect to a, b, c respectively, we have Z c ° _ dx ιγc 0θ (α + 2bx + cae 2 ) 2 ” 2(αc - fe 2 )⅛ [
∕^ + ∞ x dx nτb J- β > (a + 2bx + c® 2 ) 2 ” - 2(ac-b 2 )9 ’ ∕~ + °° x 2 dx ira J -oo ( α + 2bx + cx i } i ~ <2j ac - ⅛∙2)i t ·
Hence ∕*+ ao (⅞'+ 2¾⅛ + c⅛ 2 )⅛ τr{ac' + a'c-2bb , ') J-∞ (a + 2bx+cx 2 ) 2 2(ac-b 2 jl '
A number of other definite integrals can be immediately deduced from these by successive differentiation.
∕ a0 dx 1
^ s = —ï »
(a + 2bx+cx 2 V ha* where h≈Jac + b, we get, by differentiation, . ∕^° 0 dx _2h-b Jo (a + 2bx+cx 2 )% 3h 2 a% f™ xdx 1
Jo (α + 2δ≈ + cxj ~ 3h 2 ai ’ ∕∙= o x 2 dx = 1
Jo (a + 2bx + cx i 'ft 3Λ 2 ci
In like manner, if the equation
π
∕*2^ dx ¾∙ Jo α 2 cos¾÷j8 2 sin 2 ® “2αβ be differentiated with respect to a and β respectively, we get
7Γ
∕^2 COS‰fo ’ π
(α 2 cos 2 ® ÷ β 2 sin 2 ®) 2 4α 3 j8 ’
7Γ
∕*^2 sin¾cfe _ π
Jo (a 2 cos¾ + )3 2 sin⅛) 2 4aβ 3 '
Hence, by addition,
∕*7 dx ^ ir(a 2 ÷ fl 2 )
Jo (a 2 cos 2 a+0 2 sin 2 ®) 2 ” 4a 3 j8 3
From these other definite integrals can be readily found by further differentiation.
147. When the limits are functions of the parameter α, a definite integral admits of differentiation in like manner. For, let ∆α, Ab be the changes in the limits corresponding to the increment ∆α in a, then
Δΐί = /* 6+Δδ Φ(x, a + ∆a) dx - ∕^ b φ{x, a)dx Ja- ∖ -Δa ja ∕~b f ) ∕*δ÷∆δ
=/ γ!>{x, a + Aa)-φ{x, a)^ dx+j b φ{x, a+Aa}dx
∕-a÷Δa - ∕ φ{x, a + Aa)dx. ja
Hence, proceeding to the limit, we get du Γb dφ{χ a) , , , z , . db . da -Γ=∕ - -, — dx + φ(b, a) - τ - -φ(a, a) - τ - · da ja da ’ ' da r ' ’ ' da Integration under the Sign of Integration.
148. We shall next consider the corresponding process called integration under the sign of integration.
Suppose f b φ{x, a}dx to be represented by u, then J a j a [Jφ(x, a)daJdx-^ Φ( x > a)dx = u, or [jφtχ, α)dα]dic= Juda, the same limits for a being taken in both integrals.
Suppose a 1 and α 0 to represent the limiting values of α, then the preceding result may be written j^ a Q∕" ≠(≈> rf ∣ da ∖ dx=j' Φ( x > a)<føjda; or, adopting the usual notation, ∕" 6 ∕* φ(x, a)dadx=f i φ(x, a)dxda. Ja Ja 0 J<u t Ja
Such expressions are called double integrals, and the result just established is equivalent to the statement that in a double integration, when the limits are independent one of the other, we may effect the integrations in either order without altering the result.
It is easily seen that the preceding statement does not hold if either φ(x, a), ^ ig, a ∖ 0 r a Y. become infinite within the ’ dx ’ da. assigned limits of integration.
By aid of this principle, from a definite integral involving constant parameters we can often obtain others by the method of integration with respect to one of its parameters.
(1) For example, if the integral ∕* e~ ax cos bxdx≈ . a ... Jo α 2 + b 1 be integrated with respect to a between the limits a and β, we get ∕° 0 e - α *- e →* δ 2 + 0 2
√o X coste^=⅛log-^ s -.
(2) If the same equation be integrated with respect to b between the limits a. and 0, we get e~ ax sin ax <fo=tan-ι .
Jo χ a
On making a = 0 in this, it becomes
∕ 3 ° sin α≈ 7 ιr UX — - τr^ > X 2
provided a is positive.
<3) Again, if the definite integral (§ 133)
∕∙π∙ dx _ π J q l + ∕c cosa: - √i- κ ⅛ *
be integrated with respect to κ, between the limits 0 and sin α, we get Γ n (l÷ 81n tt cos ar ) ⅛. _ πα Jo COS X
(4) Next, if the equation
/1 Ί
x"dx=- - n + 1
be integrated with respect to n between the limits a and β, we get
∕^1 x& — x a j 1 1 + 3
∕ - ι αaj=∙log —— .
Jo log X 1 + a
(5) To find the value of the integral
∕* x u— e dx. Jo
If zx be substituted for x, we have, since the value of the integral plainly remains unaltered, u≈Γ° ze~ z2χ2 dx , Jo Γ e~ t2 ^ x2) zdx^ue~ z2 . Jo
Hence, integrating with respect to z between the limits 0 and ∞, we have ∕* f e~ * z < 1 ÷ χ2 ) zd≈dx=u f c~ z2 dz≈u i . Jo Jo Jo
A · F — Z 2 (l+x 2 ) j 1 1
Again, y 0 e >zrfz = ⅜ 1 - j , „ , ∕* x, dx 2 ιr
• · “¾ ι+≡ i ’ 4 ·
∕*0O 2 _
Consequently tt = ⅛√π; l∙ e ∙∙>J e~ x dx'=⅛ ∖ h ∣ r .
149. In many cases an unknown integral can be reduced to an elementary form by differentiation under the sign of integration.
For example, let
∕*1 tan - *(ax) , u≈ ∕ — , dx; Jo ar√l - x a then ~≈ P = =-ζ--X=r , (Ex. 5, § 113).
d a Jo (1+ α⅛ 2 )√l-z 2 2 √ι + α≡
π ∕" da τr —
∙ , u ^2√ √Γ⅛^^2 lo 8<' i + √ 1 + α '>'
No constant is added, since «=0 when α = 0.
150. A modification of this method of determining definite integrals is founded on the transformation of a simple integral into a double integral, and the inversion of the order of integration.
(1) For example, when x is positive, we have
1 ∕^° 0 — =∕ e~ χ ι>dτj , x Jo H ο sin ax . f cο . , f* ’
.'. ∕ ciz=∕ sιnαxαα ’ ∕ c~ x, Jdy Jo χ Jo Jo =f dy f c- χ us>n ∖ ax dx≈ f ~yτ¼ = v > (§148).
Jo Jo Jo a i + y i 2 va '
(2) Next, to find the value of
7Γ
∕*ΙΓ i ∕ α + δ sin 0 ∖ dθ u≈i log;; - ) — .
Jo ° ∖ a-b sin 0 ∕ sin 0
Hero, from the elementary equation 1 m + n ∕ 4 2mndx , log =∕ - ., Where m > n , m - n Jo m i - n i x i
, ∕α + δsin0∖ ∕" 1 2uδsin0<fo' we ≡ et ⅛(.-⅜a Γ>H α∙-⅜⅛⅛⅝ ;
ο i ∕*~2^ ∕* 1 dxd9 U “jo Jo a 2 - δ⅛ 8 sin 2 0
Hence, inverting the order of integration, we get (§ 133)
,∕" 1 dx . fb ∖ u≡τd> — ==ιr sin* 1 — .
Jo ∖ Ja 2 -b' i x i ∖ a J
Similarly we get
∕^2^ 1 ∕ α + δ sin 0 ∖ . ... a - √α a - b i 1 °g r~- —λ^ sin θdθ=ιr -4
o ∖ α - δ sin 0 ∕ b
Ex. 3. Again, by aid of the equation 1 ∕δ . .∖ ∕^ 1 absinθdx tan~ 1 —sιn0)=∕ ., n , ∖ a J Jo a? + δ¾ 2 sιn 2 0
it is readily seen that
7Γ
Γ 2 , ∕ b . ∖ . , τr x∕α 2 + b' i -a
∕ <tan* 1 ( —sin 0 sin θdθ= - — , . Jo ∖ a J 2 b
151. Lagrange's Theorem. — That Lagrange’s scries (§ 56) can be established by the integral calculus, and its remainder after any number of terms exhibited in the form of a definite integral, was shown by M. Popoff (Comptes rendus, 1861). His demonstration has been transformed into a simple shape by Μ. Zolotareff, in the following manner.
Let z=a2 + y≠(z), and suppose the definite integral Z {yφ{u) + x- u} n F ’ (u)dzι represented by s n , then λve get by differentiatiön = nJ" {yφ(u) +x-u] n ~ 1 F ’ (u)du-y n [Φ(x)] n F'(se) =ns n .ιy n {φ(x)} n F , (x ).
If we make n = l, we have s 0 = yφ(x)F ∖ x) + ⅛; cιx but s 0 = F(z)-F(aj);
.∙. F(z) = F(αr) + 7∕≠(ir)F'(i r) + ⅛.
ax
In like manner, if n=2, we get 2s 1 ^y 2 {φ(x)} 2 F ∖ x) + ^ i
Consequently
F«- F(x)+ li ≠m*)+⅛+[{≠(*)}W)]+ 1 ⅛ ⅛- ·
Again ss .⅛{≠(≈)} s F'(x) + -l-⅛,
. 2-⅛.jt.∕'+yΓ{≠(x)}<rte)Ι +i -l^¾,
•· 1.2 dx i 1.2.3\d®/ LÎ i J 1∙2.3<‰ 3 and so on.
Hence we deduca finally F(≈)-FM + i≠MΓ'M + i ¾ ⅛[{≠(≈)}≈F'(√]
+ · · · + E2⅛ (⅛)"∕‘ { ≡ ’ ≠<" 1+ x ^ u }" f ^ u
This is Lagrange’s series,—in which the remainder after n terms is exhibited in the form
∣⅛(⅛)y ·
Discontinuous Integrals.
152. The integral calculus furnishes many examples of discontinuous functions. For example
∕* x sin ax cos bx , , ∕"° 0 sin (a + b y )x 1 , ∕* c0 sin (a - b)x, ∕ dx = l 2 ∕ 1 — d⅛ + ⅜∕ i — —dx.
∕θ X JO X JO ae
When a + b and a - b are both positive, each of the latter integrals
(§ 148) is equal to . Hence we have—
„ ι t ∕* 3 ° sin ax cos bx , ∙π when a > b, ∕ dx = — ,
√0 ® 2
and when a <b , Γ sin α * cos ⅛ ≈0 .
Jo x
If a=b, the value of the integral becomes ~ .
4
Here we have an example of a function of two variables a and b, changing its value suddenly when b, varying in a continuous manner, becomes equal to or greater than a. This singularity has been ingeniously utilized for the purpose of obtaining the values of certain definite integrals. For example, let
∕* 0 ° sin ax cos bx. u== ∕ dx ; JO X then, since u=0 when a is less than b, and when a>b, we ’ 2
have ∕∙°t> ∕-∞ ∕ ιιe~ a da=— ∕ e^ a da≈- e^ δ , Jo 2√∂ 2
or Γ Γ g ^* d ° l ∞ c ° sfa <⅝ κ ⅛-⅜⅜->. JO JO X 2
But Γ e~ a sin ax da≈- , (§134); Jo l + af i
. ∕" 0 ° cos bx 1 ιr , h "J T+^=-2 e ·
Again, considering b as variable, ∕^ δ ,∕a ∕" c ° Z 6 sin ax cos bx , „ ∕"° 0 sin ax sin bx , ∕ ndb=∕ ∕ - dxdb=∕ . 2 -dx. JO JO JO X Jo «
Hence, if b<a, we have
∕^ c0 sin ax sin bx , τr,
∕ ä dx=-b ; Jo x 2
•ci..,. ∕" c0 si∏ ax sin bx , π if b > a, we have ∕ 5 dx= ^a,. JO x i 2
Consequently f - sin - ^ ai J in ⅛ x dx is equal to ∙∑∙ multiplied by the •z0 ae smaller of the numbers a and b.
Again, let us consider the definite integral ∕* c0 (a-b cos ic)<½κ
Jo a 2 - 2ab cos x + b' 2 '
Here we have ∕" (a-b cos x)dx 1 /V α 2 -⅞ 2 ∖
J a i -2ab cos x+ b a "*2a J ∖ + a i - 2ab cos x + δ 2 ∕ ·
_ a: α 2 - ¾ 2 ∕* dx
2α 2« J a i -2ab cos x + b 2 ’
Again (Ex. 8, § 113), ∕ dx 2 , Ha+b . x ∖ . a 3 - 2ab cos x+b' j α 2 - δ 2 an (a - b an 2 ) ’
. a a -b a f dx 1 + 1 ∕α + δ 4 , ae ∖
- ’ 2α√ α 2 - 2ab cos a + ⅛ 2 ~ a tan ^2^ j
Accordingly, if α 2 >δ 2 , α ⅞-⅞8 rπ∙ dx r
2α Jo α 2 -2αδ cos z + δ 2 ~ 2a ’
If α 2 <δ 2 , we have α ⅝ _ ⅞2 ∕- π ∙ dx _ τr i λaJo a 3 - 2ab cos x+lJ"* 2a ‘
Consequently, when α 2 - δ 2 > 0, Γ π (a-b cos x}dx . JQ a? - 2ab cos x + ⅛ 2 a when α 2 -δ 2 <0, ∕" τr {a-b cos x)dx JQ a i -2ab cos x+b 2 ~ ’ and when a=b, f π α(l - cos x}dx = 1 Γ π dx ≈ π - Jo 2α 2 (l-cosaj) 2α¾
The comparison of these three cases shows that if b be supposed to vary in a continuous manner from a value less than a to a value greater than α, the integral ∕* π ^ {a~b cos x}dx JQ d i - 2ab cos x+b i will assume for b = a-h, a, a+h the values — , 7L , 0. It is a 2a accordingly a discontinuous function.
Eulerian Integrals.
153. The following definite integrals, ∕-l z -∞
∕ α ra ^ 1 (l -x} n ~^ i dx , ∕ e- χ x^-' i dx , Jo Jo were first studied, under a modified form, by Euler, who devoted several memoirs to the investigation of their properties. They were afterwards fully discussed by Legendre, by whom they were styled Eulerian integrals of the first and second species respectively. The latter integral is now regarded as the fundamental one, to which the other is reducible, as shall be presently shown.
In the case where n is an integer we plainly have √*co ∕ e^ x X n ~ 1 dx≈1.2.3 . . . n = ∣7i.
Jo
The integral is in all cases a function of n ; and, when n is fractional, it is regarded as a distinct transcendental function. It was distinguished by the symbol Γ by Legendre, thus :—
Γ(η)=/” e^j e ai n ^ 1 dx. Jo
This is now usually called the Gamτna-Function, but sometimes, however, the Factorial Function, a name suggested by Arbogast, and subsequently adopted by Kramp and others.
Moreover, since (Ex. 6, § 115), ∕ αo z-oo e~ x x5 n dx≈n ∣ e- χ x! fi ~ λ dx, Jo we have Γ(n + l) = nΓ(n)∙
This may be taken as the fundamental property of gammafunctions, and by aid of it the calculation of all such functions is reduced to the case where the parameter n is comprised between any two consecutive integers. The values of Γ(n), or rather of log Γ(n), were tabulated to twelve decimal places by Legendre in his Traité des fonctions elliptiques, tome 2, ch. 16, corresponding to values of n increasing by intervals of ‘001 between the integers 1 and 2.
It may be remarked that Γ(l) = 1, Γ(0) = ∞, Γ( - n) = oo , n being an integer. For negative values of n, not being integers, the function has a finite value.
The first Eulerian integral, ∕∙1 ∕ X m ~ 1 (l -x) n ~ 1 dx, J0 is evidently a function of its two parameters, m and n. Following Binet we shall represent the integral by the notation B(τn, ri).
It is readily seen that / 1 z -l ic m-1 (l -x) n ~ 1 dx≈h α! n ~ 1 (l - x) n ^ 1 dx, .∙. B(m, n) = B(n, m).
Its value, when either m or n is a positive integer, can be immediately found. For, suppose n a positive integer, then (§ 123) we have
√ 1 n-l ∕∙1
j ∕ ^- ∖ l-xγ-2 dx .
By successive applications we get
∕* 1 ic rn-l∩ _ rΛn-ldχ= (w ~.I~-2)-∙ , - t . 1 J 1 χ m ~^dx JQ x μ '' , ax (m + n-lKm + n~2f.Jm + l)J 0 x
1.2.8 . . . (⅞-l)
i °° iw(∞ + l)(wι + 2). . . (rn+n- 1) The corresponding result when m is an integer is obtained by interchanging the letters rn and n.
We now proceed to show that B(τro, ri) can in all cases be expressed in gamma-functions. For if we substitute zx for x in the equation Γ(n)=∕" e- χ 3ti n - 1 dx, Jo s*∞ we get Γ(to) = ∕ e - ≈ c z n x'' ~ l dx. Jo
Hence Γ(∕ι)e- ’ ^ n ^ 1 = ∕* e-∏ 1 +*)«“+“- 1 a n - 1 <⅛B.
Jo .∙. Γ(to) ∕~ c~ t z m ~ 1 dz≈ f ∕~ e-*i 1 +*)z m + n ~ 1 x n - 1 dxdz. Jo Jo Jo
Let a(l + ae) = y, and we get
∕*° 0 β-41+x)^+"-l^ = ^ ∕^° 0 fl -y÷n-lrfy, .
y 0 (l+iε) m +"√0 (l+ar) m + n
Hence ⅛)Γ(to) c ∕-∞ fl *-⅜fo < Γ(tto + to) √ 0 (l+α,∙) m +' ’
Next let u=~ ,
1 + ae
∕-∞ a ∙n-l z -l and Z (l+¾)"÷^ aι ~Z w n - 1 (l-u) m ^ 1 du=B(nι, n).
.∙. B⅛n)-ΓwW.
Γ(jzi + to)
This fundamental relation is due to Euler.
Again, if rn=l -n, we get from the preceding Γ(to)Γ(1 -to)=∕°° ^l d χ , (§ 144).
Jo 1 + x sin nτ
If to=⅜, this becomes Γ(⅜) = Vπ.
This result agrees with § 148, for, if we write z 2 for x, T(i)≈f ∖ -*χ-ldx≈2∕' a, e-* s dz. Jo Jo
154. Many definite integrals are reducible to gamma-functions, of which a few elementary cases are here given.
(1) To express the definite integral
7Γ
∕ 2 sm n ~ 1 θ cos m ~ 1 θdθ Jo in gamma-functions.
Let x∙=sin 2 θ, and the integral transforms into
,1 « 1 ~ 1 r (⅜) r (⅜) i Z < 1 → τ < fe -* -ψ⅛J- ·
(2) To find the value of ∕ x m (l-x n ) r dx. Jo
Let x n =z, and the transformed integral is
1 r 1 ∙=±l. 1 , 1 ... 1 r (≡i i ) r < 1 ∙÷ 1 > σ ^* i r(, +l ÷≡±-*)
(3) If in the last r= - ⅜, we get ∕- τ(-)
∕∙1 x! m ~^ i dx Vτ ∖ n J Λ (i-x)i~ n r (i+ wθ ’
∕∙α
(4) u=∕ (a - x) m (x - β) n dx. Jβ
Let x≈β + (aβ)z, and we readily find
! θXrn-l→ι-l-l T(WI+ 1)Γ(to + 1) il -(.-⅜)-+-÷' ' r ( m+ √+2) ·
(5) To prove the equations g - ax χ n -1 cos l χ dχ≈ κ , . cos uθ J
(α 2 + ⅛ 2 )ι
∕* e- a3c x n ~ 1 sin bx dx≈ -- sin nθ;
-∕θ (α 2 + δ 2 )ι
where 0—tan -1 (—.
∖ α )
In the equation f e -αz ic n-l i fo al .∏.⅛ , Jo a n let a - ib be substituted for a, and we get Γ a nr , 1 , . · , „ 1 , Γ(to) Γ(n)(α+iδ)-
If b≈a tan θ, we have α . b . λ
,-<, , ,,τι — cos θ, ≡—τ≡rτ ≡ sin θ.
(α 2 + δ 2 )t ’ (α 2 + δ 2 )i . '. f e ^ <∞(cos bx + i sin bx)x n ~ 1 dx = -⅞1— (cos nθ + i sin nθ).
* 7 ° (α 2 + δ 3 )i
Hence the proposed equations are obtained by equating the real and imaginary parts respectively.
(6) f cos bx x n -~ l dx=J^ ny, cos —; Jo b n 2
∕* siu bx x n ~ λ dx*J-f i - gin — . √0 l> n 2
These follow from the preceding by making to = 0.
A more rigorous demonstration of this and of the preceding example will be found in Serret’s Calcul intégral, pp. 194-198.
(7) Find the value of w-r(—W—W—)... r(—).
Here, since Γ (— )r (1 ) = -dL- ,
∖ n ∕ X n ∕ s j n 2L n it is easily seen that 2- π n ~ 1
. π . 2π . (to-1)tγ sin — sin — ... sin —
η η n
But it can be shown, by trigonometry, that . ιγ . 2ιr . (to-1)iγ to , sin — sin — ... sin i —- i - = o --, » to η n 2 n ~ l hence r μ')rU)..∙Γt--l-¾) 4i ∙ ∖ TO J ∖ TO ∕ n TO’
(8) Prove that
ιr
∕" 2 cos n 0 cos mβ dθ; —-. Γ(ra + 1) ,
√ 0 2 n ÷ 1 r (’ ⅛2 + l)r(li2-l)
155. We next proceed to show that Γ(w) admits of being exhibited as the limit of the product of an infinite number of factors, a form which was adopted by Gauss as the definition of the function.
If in the equation Γ(to)=∕ g-¾ ,, -Mic Jo
λve make e* ac =>i, we get Γ(to) l°g ⅜) t ^ 2 ‘
But (§ 63, Ex. 9) log — is the limit of μ(f - z^ when μ increases beyond limit.
.∙. Γ(TO) = lim. μ n ^^∕*(l-2*^') n ⅛
=1im. μ n ∕' l y μ - 1 (l-yf-' l dy, making z≈y> i . Jo r(TO) = lim. /i” · ! , Λ∖ 3 '- 7-⅞-> ’ ( § 123) ’ n . (to + 1) . . . {n+μ) when μ is increased indefinitely.
As an application of this definition of Γ(to) suppose n + l and to - l respectively substituted for to, and we readily obtain
{Γ⅛)} ’ _A _ P Vi Yι M
Γ(n-i) Γ(π + !) ∖ " ’ Λ ("+l) s Λ (» + 2)V' '
to . lτ ∙= - r - sin — » lT TO by a well-known trigonometrical relation.
If we make to=1, this gives
Γ(l - Z)Γ(1 + 1) - -⅛-; . ∙. Γ(Z)Γ(l - Z) =≈ √L , as before, sin lτ sin lτ
156. Again, if we make x=az, we get
∕* α
∕ ae”* ^ 1 (α - x} n ~ 1 dx ∙≡ a m + n ~ 1 ∕ z m - 1 (1 ~ z) n ~ 1 dz .. am . n . 1 Γ(⅛ΓW
α Γ(7TO + TO) This result can be readily represented as a theorem in double integration, as follows.
If the double integral ffx m H) n - λ dxdy be taken for all positive values of x and y subject to the condition x+y<a, its value is represented by r (m)Γ(τι) oe „, + „
T(»l+ 71+ 1)
For, considering x as constant, and integrating with respect to y between the limits 0 and a-x, the value of the double integral becomes
1 fa a m + n Γ(τn)Γ(7i + l)
-J a x∙*- ∖ a-xr<lx, or —
i , e ,,
Γ(77⅛ + Tl+1)
157. The preceding result, first given by Euler, was generalized by Dirichlet (Lionville’s Journal, 1839), and extended to a large class of multiple integrals by the following theorem.
Let V = j^J~J~· · λ y m - χ zP-' i . . . dxdydz . . . , in which the variables x, y, z, &c., are always positive, and subject only to the condition then y c . τ I(0Γ(τn)Γ(n). ’.
It will be sufficient here to show that the theorem is true in the case of three variables, x, y, z ; ι.e., let
V "J~J'j' jji HJ m ~ 1 tf , ∙~ 1 dxdydz, subject to the foregoing conditions.
Integrating with respect to z, considering x and y constants, we get
V = ^JJx l - ∖ ∕ n - ∖ X - x —y) n dxdy , in which x and y are positive, and subject to the condition x + y< 1.
If we next integrate with respect to y, between the limits 0 and 1 - x, we have
∕* 1 V* -1 (i - « - y) n dy≈ (1 - aj) m + n Γ ft w ) r Q ι + l} e
√o Γ(7n + n + l)
Accordingly
V = -∏ ,n ⅛L ∕^ 1 α∕ -1(1 - χ'y^+*dx Γ(m + M + l∣∕ 0 k , Γ(w⅜)Γ(⅛) Γ(Z )Γ(τ⅞+⅞+l) Γ(Z)Γ(τn)Γ(^)
Γ(τn + 7i + l) Γ(Z + 7⅛ + n + l Γ(Z+ττι+n + l) *
158. The preceding theorem when extended to n variables can be stated somewhat more generally, viz., if
· · ≈ z -iy ,n -⅛ n -ι . . . dxdydz . . . where x, y, z are always positive and subject to the condition ≡÷(⅜) ’ ÷(y)'÷--.
a, β, y, . . . ρ, q, r, . . . being positive quantities, then we shall have γ α z 3 m γ" . . r G^) γ ( 7 ) γ (t)∙ · *
^∙.∙. r Λ + ± + ^ + h ...) ∖ p q r )
This readily follows from the preceding by making
(⅞y^- ⅛ρ- (fp-∙
In the case of three variables this theorem contains a large number of results relative to volumes, centres of gravity, moments of inertia, &c.
The remarkable elegance and generality of Dirichlet’s theorem immediately attracted notice, and his results were speedily extended by Lionville, Catalan, Leslie Ellis, and other mathematicians of distinction. Of the results thus established we shall content ourselves with giving Lionville’s extension of Dirichlet’s theorem (Liouvilles Journal, 1839).
If v 3^∙∙ a = ,-1 ^^ 12n ^ 1 ∙∙∙∕ J(⅞) p +(⅜) , ∙∙∙5 cω ^∙∙∙> where x, y, z are always positive and subject to the condition
(l)'÷(⅜)'÷(y)'∙∙< i ∙
a, β> Pt B being always positive, as before, then will v _ α*∕3 m γ n . . ∖ p J ∖ g 7 · · · /·» ⅛,,. ., P<B... γ (L + TL jjo w '' * fWu. ∖ p + 1 )
This follows without difficulty from the preceding by assuming (^~y= ic ' > · · · » and then making √ + y , + √... =u.
A few examples are added for illustration.
(1) The value of fff dx 1 dx 2 dx 3 , . . dx n , where x 1 , x 2 . . . x n are subject to the condition x l 2 + αr∣+≈ 2 . . . + a⅞ξ<R 2 ,
⅛ ≥ ·
2 " r (1+ ⅜)
(2) The value of ΓΓΓ dx i dx, 2 . . . dx n JJJ √1 -xf-x]... -xf t ’ extended to all positive values of the variables for which the expression is real, is
»+1 7Γ 2 2 ^ r (⅛ 1 )
(3) The value of jj^ xh-~ l y- k e x +^dxdy extended to all positive values for which x + y<h is -⅛(e*-l). sin kιr
(4) The value of for all real values of the expression, x and y being positive, is XH∙
(5) The value of JJJχP ~ 1 yi - 1 z r - 1 dxdydz extended to all positive values of z, y, z contained within the ellipsoid
α⅞. e - r (⅜) r (⅜) r (⅛)
18 8 r(ε±⅛% 1 )
(6) Prove that
∕y . . Λ⅜÷-⅜∙∙∙÷^⅛^ ... dxnι JJJ √1 - x ∖ - X i . . . - x n when extended to all values subject to the condition x* l +x* . . .+≈, 2 ,<1, πϊ /Ί
is equal to Λ^≈ c )(l -α *) 7 dx, where k=^fa[ + α 2 + . . . + α, 2 ,.
159. We shall jnext give a short account of Legendre’s formula for the calculation of log Γ(l + x).
Adopting Gauss’s definition, substituting x +1 for n, and taking the logarithms of both sides of the equation of § 155, we get log r(ae + l)
=lim.∣αj log μ-log (1+ f)- 1 °g (1 + ⅞)∙ · · -lθg (1 + 7^)l '
If now x lie between +1 and -1, we may substitute their well-known expansions for log Ç 1 + ∣ > l°g ^1 ÷ . . .
Hence, representing the indefinite series
⅛ + ⅛ + ⅛ + · · · &c ·
by s n , we shall have log Γ(x+1)=-γx + ⅜s 2 x 2 -⅛.9 3 x 3 + ∣s 4 x 4 -. . . &c., where γ represents the limit of
τ + i + ⅛+ · · · + — - l°g M
when μ is indefinitely increased. This limit, whose importance was first noticed by Euler {Acta Petropolis, 1734), is now usually called Euler's Constant.
If we change the sign of x in the preceding equation it becomes log Γ(l - x) = yx + ⅜s 2 x 2 + \s. A a? + ∣s 4 x 4 + &c.
Λσain Γ(1+χ)_ {Γ(l+x)} 2 [Γ(1 + x)] 2 sin xπ E Γ(l - xj~xΓ(x)Γ(l - x) “xπ
Hence λve have r l'Ιι log Γ(l + x) = ⅜log -T—--(7x+½x 3 + ⅜s 5 x 5 + . . .).
>3 III <4z7Γ
Again, by logarithms,
Σ *f^ iC
⅜ log j-3^=αr+⅞x 3 + ⅜x 5 + fcc.
Consequently log Γ(l⅛x)=⅛ log 1 w(∣⅛)+(l-γ)af-i(¾-l)x s -i(⅛-l)χβ.. .
, 1 xπ , 1 1 +x , ,
=⅛ log ^iΓ^-* log iτ^ +c ι χ - c ^-⅛ x5 ∙ · ·, bill «/>71 1 ∙D ’ where c 1 = l-7, c 3 =⅜(¾-l), c 5 =⅜(⅞-l) . . .
It is easily seen that the constants c 3 , c 5 , &c., form a rapidly decreasing series, in which each term can be calculated to any required number of places of decimals. Accordingly, when the value of Euler’s constant 7 has been determined, a series of values of log Γ(l + x) can be computed from the foregoing equation, and thus tabulated.
Again, since Γ (-χ-) = Zr^> the value of 7 may be calculated by making x=⅜ in the preceding formula; by this means its value is found to be 0'5772156649 to ten decimal places.
On the Integrals Li x, Ei x, Si x, and Ci x.
160. Having thus arrived at a determination of Euler’s constant, we shall return to the consideration of the logarithmic integral and other transcendents introduced into § 131.
Adopting the notation of that article, we have p x e~ z f l e~ xu
Li(e^ x ) = ∕ — dz = ∕ du, writing xu for z; a ∕ 25 ∕ Vj
αo ∙ z 00
∕∙∞ p - yu _ p - xu
. ·. Li(e-≈) - Li(e-*) =∕ - — du Vi u f a >c-v u -e- χu , f l c- χu -e~v u , =∕ du+∕ du. Jo U -∕ o u
But (§ 142), z -∞ e -yu- e -xu χ ∕ du = log —:
√o u y TA 1 ∩ - yu ∕A 1 _ p - χu
.∙. Li(e - χ )-Li{e~v') = log x- logy +∕ '—du·/ ——du. Vo u V a u
. · ∕* l l-e-v u , ∕∙ 1 1 - (1 - u)y , ∕' 1 e-v u -(l-u)y 1
Again ∕ du≈ I ’ -du-∕ - -du. JO U √ 0 U J o U
But ∕*l≤⅛√ ’ ⅜S⅛⅛ i
Λ u Jo 1-(!-«)
=∕ {l + (l-u) + (l-w) 2 + . . . +(l-κ>-ιl⅜
∙ z 0 ’
=ι+⅛+i+ · · · +—
y
If now we suppose y to increase beyond limit, observing that in that case Lie*»=0, and that 7=limit of l+⅜+⅛+. . . 4-1-logy when y-∞, we get
Ui i -∙)- τ+ logs√ 1 lz<¾ 11 . ltol A->--(i-u)⅛ √o u Jo u
We next proceed to show that -u)y, ∕ - -du Jo u vanishes when y becomes infinitely great.
To prove this, we observe that, since u lies between 0 and 1, e~ u >l-u, .∙. e~y u >(l-u)v.
Also e u >l+w, .∙. (1 - ιz)e">l - u 2 , hence 1 - (1 - w)»e»“ < 1 - (1 - u 2 )* < yu*, .∙. e-v u -(l-u)y<yu 2 e- u v.
Consequently f e ^~^~ u ^du<yJ ue- u vdu. Vo u Vo A J
Again, yf ue~ u vdu∙= — (1 — e~ v )- e _ »=0 when y=∞. Vo y
∕∙ 1 e-* u -(l-u>, • . ∕ (Jib Jo u vanishes at the same time.
Hence Li(e-«)=.7 + logx-/* 1 ~ e ^ du. . . Vo u
Again ∕" 1 l-e- itu . ∕^ 1 x 2 √ ’ ∕ — u du ^∕ ( x ~ Γ~o u + Γ~0~V'' i - · · · y > dii J θ u ÷∕q 1.2 1.2.o
-*-h⅞ + ⅛⅛r ft0 ∙
rγΛ λ∙>3
Li(e-≡)≈γ+ logx-x + ∣- - ⅜ + ⅛c,
1.2 1.2.0
Again, Li(e*)=∕* - *- = -Γ-dz V aa z V—x z -->4Z^ , ^÷Γ⅛λ ∣ when c vanishes.
∕ ∞ ∕-∞ pX
'°∕ +∕; JX J€
.∙. L·(..)- -ι in >.
∕-⅞i- e → ∕* α ⅛→, =∕ dz-∕ —dz. Jo z Jχ z
Let 2=>xw, and this becomes ∕Ι z >ru _ „ - zu ∕∙ ∞ p - xu lΛ{e x }≈=∕ du-∕ —-du; JO u J ∖ “ /*!<>»« _ e-xu
.∙. Li(e x )∙=Li(e- χ ) + ∕ du ∖
, ∙×o w hence
Li(e x ) = 7 + l 0 gx + χ + ⅛J≤- + ⅜ θ + ⅛c∙ 1.2 1.2.0
This and the preceding can be represented by the single formuh
Eix = Li(e x ) = γ +1 log (x<) + χ + + ⅜ ⅛ + &c.
1 · 2 1.2.0
The expansion for the sine-integral can bo readily obtained, fo we have by definition σ . ∕* τ sin z , Sιx=∕ dz: Jo ? hence, substituting the ordinary expansion for sin z, and integral ing between the limits proposed, we get Q∙>3 zy45 8i ∙- , ^⅛÷⅛ i .u^ to ∙
Again, if, in the equation already proved / I· P ~ XU ~du = y + i log (χi)- x + ⅛- - ⅜ __ + &c., we substitute ix for x i it becomes / 1 P “iX^ <γti ∣ t ∕y∙3 J—du = y +11 og (x< ) - ix - ⅛ + ⅛ ⅛ + & c.,
∕* 1 cos xu - i sin xu j t 1 1 ,a x . t „ or ∕ du= y +1 log (x 4 ) - ix - &c.
y oo u
Hence, equating the real parts on both sides, we get
∕ 1 cos xu, 1 , 1 ... 1 x 2 1 . x 4 -- r du^y + l log (x*)- ⅛ rl + έ r aχ i -
Consequently
Cix = γ + i log (x<)-⅜ ⅛ + ∣--&c. 1.2 1.2.0. i x
The several series here arrived at are readily seen to be con vergent for all real values of x, and by aid of them the values ol Eix, Six, Cix for different values of the argument x can be tabulated.
Such tables have been constructed by Soldner, Bidone, BretSchneider, Schlömilch, and others. The most recent and complete tables are those of Mr J. W. L. Glaisher, already referred to (§131).
161. The values of some definite integrals can be best determined by transforming them into infinite scries. This statement will be illustrated by one or two examples.
(1) To find ∕^ 1 ⅛ 5 <fos.
Here, when x is less than unity, l2KΞ= logx(l+∞ + ar ’ + . . .+x"+ &c.); 1 ~ 35
∕∙1 1
but ∕ X n log X dx≈ - 7= ra; Jo (l+n) 2 consequently
Z 1 lθgO! , ∕ 1 1 1 π 2 Jo ι⅛ ito =-(1 + i + ⅜ + · · ·) = 6'
(2) In like manner it can be shown that
∕* 1 loga! π 2 ∕* 1 logaJ , τr 2
√0 1 + ae 12 √ 0 1 - ae 2 8
(3) Again, to find ∕~ 1 1 ~ x ,J c l x ,
√o (l+ae)logae
Replacing -2— by its development, we get 1 4* 35 χa ≈x a ~ 1 -x~ a - (x a - x 1 ~ a )+x a + 1 -x 2 ~ a - (x a + 2 -x 3 ~ a ) . . .
Consequently (Ex. 4, § 148)
∕-l x a-ι~ x -a α a + l α + 2
∕ -rτ~. —n <∞= lθg ï log h log . . .
JO (l + ic)log∞ 6 l-α o 2-α β 3-α
= κ~ α ( a + 2 )(≈ + 4 ) · · · (2-α)(4-α) . , , ë (1-α)(l+α)(3-α)(3+ α) . . .
= α(2 2 -α 2 )(4 2 -α 2 ). , = 1 t ar o (l 2 -α 2 )(3 2 -α 2 ). . . g 2 ’
by a known formula in trigonometry.
162. Conversely, an infinite series can in many cases be transformed into a definite integral, and thus evaluated.
For example, suppose
S = l+⅜-⅜-⅜∙+⅜+τ⅛ -&c.
1 r 1
Here, since = ∕ ae 2n dae, we have
2n + l√ 0 S≈∕' 1 dx(l + x 2 -x i -a^ + . . .)=∕ 1 A±≡⅛=2L√2.
√0 JO 1 + a ^ 4
In like manner we get
ι 1x1 i u. i ∕ 1 l-∞ 6 , πl + √2
ι-÷+i-Λ+A-∙.-^ ιZ3> lfe -τ-2-·
Again, the series 8 Xp + 1) · · · (p + «·) + (p + «i)(p + rø + l) . . . (p+wι + n) + ^ c ' can be represented by a definite integral.
Here - p(p + l). . . (2> + n) - Γ(n+2> + l) - Γ(n + 1)^* XP dX
(§ 153);
1 ∕* 1 ·’· s = Γ(π + l)¾ (l-α) n (≈ p ^ 1 + ≈ m+p - 1 + · · · )dx 1 r 1 (i-¾)⅞p-^
1.2.8... 7√ 0 i-»™
We now proceed to give a few applications of the calculus to geometrical problems.
Areas of Plane Curves.
163. If a plane curve be referred to rectangular axes of coordinates, the area between the curve, the axis of X, and two ordinates corresponding to the abscissae a and b is represented by the definite integral ∕ b ydx.
Hence if y ≈φ(x) be the equation of the curve, the area in question is denoted by ∕ b φ{x}dx .
From this result it follows that every definite integral may be represented by an area. And it is seen at once that all the examples hitherto considered admit of geometrical interpretation.
In the above formula the ordinate is supposed positive for all points of the curve between the limiting abscissae. The modification when the curve cuts the axis of x can be yeadily supplied.
Ex. 1. Let the curve be an ellipse, represented by the equation ^ + ^- = l α 2 + δ 2
Here y = —λ∕α 2 -x 2 ; and, if x, y be CL the coordinates of the point P (fig. 7), the area APN is represented by the integral b r a
— ∕ V ft2 ^^ j -^'h ’ ' a Jx
' Let x≈a cos φ i and the integral transforms into abΓ⅛ shι 2 φdφ= c ^l- (φ - sin φ cos ≠) =^cos^ 1 - - .
fc ½) Z (l À
Hence, the area of the elliptic sector APCP 1 is equal to 35 ab cos -1 — · a
If the sectorial area APCP 1 be represented by S, the preceding result gives x S y . S
— = cos — r , ⅜ = sin— r ; a ab o ab x y . , S
or —=cos φ , -⅛-<≈sιn<⅛. where φ≈- τ . a b ab Ex. 2. The equation of a hyperbola referred to its axes is
α 2 lp
•'∙ 2∕ = 4√ aj2-α2 ·
Accordingly, if x, y be the coordinates of the point P on the curve (fig. 8), the area APN is represented by b rx b .- 5 - δ ab 1 x + √x 2 -α 2
- ∕ √x 2 - a 2 ⅛= - a 2 - -y log ~
∙za xυ ab . ∕ x , y ∖
-=T-2- lθg (a + fλ
Consequently the area of the hyperbolic sector AGP is represented by
⅞1 0g (≈ + ⅜).
2 ° ∖ a b)
This relation has given rise to a class of expressions called hyperbolic functions. Thus, if S denote the area of the hyperbolic sector APCP 1 , we have
≡-M⅞÷f)-
A + X. e ⅛ .
a b
Hence, from the equation
α 2 ⅛ 2 1 s we get β - 0⅛ .
a b (Ύ
Let — be represented by v, and we have ab x _ e v +e~ v y _ e v -e~ v a 2 ’ b 2
In analogy with the formulae for the ellipse the expressions e’ + e-” d ’ -e~ v
2 ’ 2
are called the hyperbolic cosine and hyperbolic sine of v respectively, and are usually written cosh v, sinh v ; and we have
— = cosh v , J- = sinh υ · a b
Again, for simplicity, the hyperbola may be assumed equilateral, and a=b = l ; in this case the equations become ae = coshv, y=sinh υ, where v represents the area of the corresponding hyperbolic sector, represented by APCP j .
Also, by analogy, wc write
, , sinh υ e*-e^ ν e 2 * ’-l , tanh υ∙= — ξ - = -—-—- = - - , &c.
cosh v e τ +e~ ν c 2 * + l
Again, we obviously have cos x =cosh ix, i sin x =sinh ix .
Between hyperbolic functions many relations exist analogous to those between ordinary trigonometrical functions.
For example, it is easily seen that we have [table]
Again
-j-sinh x— cosh x , - r cosh x== sinh x ∣ dx ’ dx d . . 1 d .. 1
- i - tanh x= — i - 5 - , - 7 - coth x≈ — r --,- 5 ∙∙ · dx cosh⅛ dx sinh*«
Hence J" :osh xdx=smh x, J^ sinh xdx=co^a x, f - ⅞ “tanh ® > f∙^⅞~ — - coth x . J cosh⅛ J sιnh⅛
The analogy between hyperbolic and trigonometrical functions will also appear as follows. If we make z=sec φ in the equation of the equilateral hyperbola x 2 -y 2 = 1, we get y=tan φ. Consequently sec φ = cosh υ. tan φ = sinh υ.
Moreover the equation v = log(z+y), gives
H⅜÷τ)-∙
In this case φ is called the hyperbolic amplitude of v ; and, by analog y, we write ≠=amh v. Also when x =cosh v, we have v=log (ae + √x a - 1). Again, when y=sinh ν, we have v = log (y + Vy 2 +1 ).
Moreover, since υ is the sector whose corresponding hyperbolic cosine is x, the connexion between v and x may be represented by the relation v=sect cosh x.
Hence we have sect cosh α=log (z + Vz 2 -1).
In like manner we get sect sinh x = log (ae+∖ z ⅛ + l), sect tanh z=⅛ log 1 +x .
1 -x
This notation exhibits the analogy between the elementary integrals in a remarkable manner, and even more strikingly when we introduce the Continental notation, arc sin x, instead of sin -1 », &c.
Thus ∕L ∕fe— —sin - ⅛=arc sin x;
√√l-x 2
/ .⅛⅛=log {x +√1+ae 2 ) = sect sinh x;
√l+χ 2 0 y ^ CL'iXj
-=tan-⅛=arc tan x; l+x 2
∕^^ ,,=⅛log l±^=sect tanh x. J l-x i 2 °l-a≈
Ex. 3. To find the area included between the cissoid of Diocles and its asymptote.
The equation of this curve is y ∖ 2a,-x)≈x ∖ and that of its asymptote is x=2a.
Hence the area in question is represented by f~ ila fidx Jg (2a-x)C
Let x=2a, siu 2 0, and the integral becomes
7Γ
8α 2 ∕" 2 sin 4 0d0 = ∣πα 2 .
√0
Ex. 4. To find the whole area of the curve
(W∙ 1 ∙
Let —=cos"ø; then-^-= sin"0.
a b
Hence the whole area is represented by ±nabΓ 2 sin n + 1 0 cos ,, ^ 1 0d0.
Jo
The method of determining this integral has beeu exhibited in § 122.
164. In the preceding examples the area of any portion of a plane may be conceived as divided into a system of infinitesimal rectangles, clxdy, by lines drawn parallel to the axes of coordinates, Accordingly any plane area may be represented by ffdxdy, taken between limits determined by the boundary of the area.
Again, as in polar coordinates, the plane may be divided by a system of circles having the origin as centre, and also by a system of radii vectores drawn through the origin. In such coordinates the element of area bounded by two circles of radii r and r + dr and by the radii vectores corresponding to the angles 0 and Ø + rfØ is plainly represented by rdrdθ. Accordingly, any plane area may be represented by ffrdrd9 taken between the limits determined by its boundary.
Hence, if the equation of a curve be given in polar coordinates, the sectorial area S bounded by two radii, and the curve is represented by
S = ⅜∕ r 2 d0, J α
where a and β are the values of 0 corresponding to the limiting radii.
(1) For example, in the class of spirals represented by r=aθ m , we have
α 2 02<n+l s ^T ⅛+ι +c ° nst
If the area be bounded by the prime vector 0=0, this gives
«2 02rn÷l
2 2≡i + l ‘
Thus for the spiral of Archimedes, whose equation is r=aθ, ≡-4*∙-≤.
In the spiral, r*=α0, we have q ~α 2 0 2 ∕r 2 ∖ 2
4 (2a,) ‘ In the reciprocal to this.spiral, viz., r s 0=α 2 , we have s = ^y log θ=α 2 log^yj , in which the sector is reckoned from 0=1.
(2) To find the area of a loop of the curve r 2 =α 2 cos nθ.
Here r =0 when nθ= ~, and r=a when «0 = 0. Consequently the area of a loop is represented by π
α 2 ∕ 2rt cos nθ dθ, Jo α 2 and, accordingly, is —. It is easily seen that when n is a positive integer, the curve consists of n loops; accordingly the entire area of the curve is α 2 .
(3) To find the area of the loop of the folium of Descartes, the equation of the curve being x i +y 3 ≈8axy.
Transforming to polar coordinates, we get
7Γ
S = — s i n2 0 co ≡ 20 d0
2√ n (sin 3 0 + cos 3 0) 2 ’
Let tan θ = u, and this becomes
9α 2 ∕* 0 ° u i du _3a 2
T√ o (l+w 3 ) i 2^*
165. If from any point a perpendicular be drawn to any tangent to a curve, the locus of the foot of the perpendicular is called the pedal of the curve with respect to the assumed origin.
If p and ω be the polar coordinates of the foot of the perpendicular, the sectorial area of the pedal curve is plainly represented by
⅜∕p‰
taken between proper limits.
The following remarkable connexion between the pedal areas with respect to the same closed curve, for different internal origins, is due to Steiner. Let A be the area of the pedal with respect to the origin 0, A ’ the area for origin O', and p, p' the corresponding perpendiculars, then we have z -2jγ ∕-2π
Λ = ⅛∕ A' = ⅜∕ p" i dω. Jo Jo If, now, x, y be the coordinates of O' with respect to a pair of rectangular axes drawn through 0, we shall have p' ≈p - X cos ω - y sin a; therefore
-2>r √-2π∙ ∕∙2π∙
A'-A≈⅜∕ (xcosω + ysin a') i da-x∕ pcosada-yf jjsin«d«.
√0 √0 Jo
But
∕-2π ∕-2π . ∕-2π
∕ cos 2 « da = π , ∕ sin 2 « da = π, ∕ sin a cos « da = 0; Jo Jo Jo consequently A' - A = γ(≈ 2 + y 2 ) - gx-hy ,
√-2τr z-2rr where g≈∕ poos a da, h=∕ ^jsinwd«. Jo Jo
Hence we infer that, if 0 be fixed, the locus of O', when the corresponding pedal area A' is constant, is a circle.
All the circles obtained by varying the pedal area are concentric. Also the common centre is the point for which the pedal area is a minimum, and the pedal area with respect to any origin exceeds the minimum pedal area by half the area of the circle whose radius is the distance between the pedal origins. Many interesting results may be deduced from this theorem. When the curve is not closed, it is easy to prove, as was shown by Prof. Raabe (Grelle, vol. 1.), that the locus of the origin for pedals of equal areas is an ellipse. The corresponding theorems for the volumes of the pedals of surfaces were investigated by Dr Hirst (Transactions of the Royal Society, 1863). In addition to other important generalizations, Dr Hirst has here proved, when the surface is closed, that the locus of the origin for equal pedal volumes is a surface of the second degree.
Another remarkable theorem of Steiner’s, on the connexion between the areas of pedals and of roulettes, may be stated here. When a closed curve rolls on a right line, the area between the right line and the roulet>e generated by any point invariably connected with the rolling curve, in a complete revolution, is double the area of the pedal of the rolling curve, taken with respect to the generating point as origin. Hence it follows that there is one point in a closed curve for which the entire area of the roulette, described in a complete revolution, is a minimum. Also, the area of the roulette described by any other point exceeds that of the minimum roulette by the area of the circle whose radius is the distance between the points.
Rectification of Curves.
166. The rectification of curves is based on the principle that the length of an arc of any curve is the limit to which the perimeter of an inscribed polygon approaches when each of its sides is conceived to diminish indefinitely.
Hence, if the curve be referred to rectangular axes of coordinates, and if ds denote the element of the arc of the curve at the point (x, y), we shall have ds 2 =dx 2 + dy* , and accordingly ∙-f ! 1+ (l) , ( ' dx > ° r s ~f i 1 + (⅞)" Γ rfs taken between the limiting points, i.e., the extremities of the arc.
In like manner if the curve be referred to polar coordinates we shall have ’ y^ j ι+ (⅛ 9 ) 1 ι ' dr ∙ ° rs -f ∖ r, + (⅛) ι s * dβ
We shall illustrate these formulae by a few simple cases.
Ex. 1. In the ordinary parabola x 2 =2py we have dx p ’ .∙. ( χ2 +P 2 ) i dx (¾ a +ff 2 )⅜ ι p ≈+√ff 2 + ¾ 2 2p + 2 1 ° g p Ex. 2. In the more general parabolic curve represented by x n ∙=py we have dy n „ 1
-" = _aj«-i; dx p
.∙. s -∕^(ι + "⅛-≈)⅛.
This expression is capable of integration in a finite algebraical form (§ 123) tor the following values of 2n - 2,
1. ⅜, ⅜, i,.. -, &c., r i.e., when η is
I, J,l,... , &C.
2r
167. In illustration of the method of rectification in polar coordinates, we commence with the spiral of Archimedes, r=aθ.
Here s =l-J ∖ r * + aSfidr .
This shows that the length of any arc of this spiral is equal to that of a corresponding arc of a parabola.
This relation between the spiral of Archimedes and the parabola was discovered, according to Sir John Leslie, by Gτegoire St Vincent, before the middle of the 17th century (see Leslie’s Geometrical Analysis, p. 424). That a corresponding relation connected the parabola y η ≈px and the spiral r n ^ 1 = 1 HLfρg was established by
71
John Bernoulli (Acta Erud., 1691).
These results were extended by Lardner (Algebraic Geometry, p. 355), and in their general form may be stated thus :—
If from the equation to any curve in rectangular coordinates 'another curve in polar coordinates be formed, by making dy = dr and dx = rdø, then the length of any arc of the second curve will be equal to that of the corresponding arc of the first curve. Also the sectorial area of the second curve will be half the area bounded by the corresponding y ordinates in the first curve.
These relations can be immediately established.
As an example, the right line y==rηx gives by this transformation the logarithmic spiral r=e mβ . Hence we can always obtain a portion of a right line equal in length to any arc of this spiral,— a result which is obvious otherwise.
Again, from the ellipse
⅛ + ⅜= 1 we 8 et ≈ = -y√⅛ 2 -y 2 ,
.∙. daj =-⅜-⅛.
b ∖ , b 2 -y 2
Hence the differential equation of the transformed curve is a dr dθ= -- i τ===^ , h ∖ Zb 2 - r i from which we get r = b cos —Θ , a where θ is measured from the line which corresponds to the major axis of the ellipse. Accordingly, the rectification and quadrature of this latter curve is the same as for the ellipse. This can also be shown immediately otherwise.
168. Whenever the pedal equation of a curve (§ 165) can be found, there is another general formula for its rectification, which may be proved thus.
In fig. 9 let ON be the perpendicular let fall on the tangent at any point P on a curve, and ON' the perpendicular on the tangent at a consecutive point Q; and suppose ON =p, angle AON = a, and PN = t. Then PQ = Δs, angle SON = ∆ω, Δi = QN'-PN. ~ nn ι∙ PT + TQ dt 1 . QN'-PN
Hence — = lιm. - , 3 -->hm.- .
da ∆ω da Δ«
But PT + TQ + PN - QN'=TN-TN';
. α ds dt V TN-TN' 1 . SN nxr hence — — = lιm. = Inn. — = ON = ρ. da da ∆ω ∆ω
Accordingly, if ω 1 and « 0 be the values of a corresponding to the extremities of the arc s, and i 1 , i 0 the corresponding values of t, we have
∕* ω ι s=i 1 -i 0 + ∕ P d <*>∙
This theorem is due to Legendre. In its application it is well t® observe, that
-⅛=lim.- = lira. TN'=i. da ∆ω
For example, in the parabola we have a e dp a sin a - d cos a ' " da cos 2 «
Hence, if s be measured from the vertex of the parabola, we have a sin a Γ" i da sin a , , ( π ι a ∖ s =-,-+α∕ a — 3 -+α logtan - r + - δ -).
cos 2 « Jo cos « cos 2 « ∖ 4 2 ∕
Similarly in the ellipse,-^ ⅛ = 1, we havej9= √α 2 cos 2 ω ÷ δ 2 sin 2 ω.
cL o
Accordingly, the rectification of the ellipse depends on the integral f α 2 cos 2 ω + δ 2 sin 2 ω da.
Likewise the rectification of the hyperbola depends on the integral J √ α 2 cos 2 w - 5 2 sin 2 w da.
Further considerations on the rectification of these curves will be found under the head of elliptic integrals.
169. Steiner’s theorem connecting the rectification of pedals and roulettes, analogous to that which connects their areas (§ 165), may be here stated. It is as follows :—
If a curve roll on a rigid line, the length of the roulette described by any point connected with the rolling curve is equal to the corresponding arc of the pedal of the rolling curve, taken with respect to the describing point as origin.
From this it is easily seen that the length of any arc of a cycloid is equal to that of a corresponding portion of a cardioid, and the length of a trochoid to an arc of a limaçon. Again, if an ellipse be supposed to roll on a right line, the length of the roulette described by either of its foci is equal to the length of the corresponding arc of the auxiliary circle.
Rectification of Curves of Double Curvature.
170. If the points in a curve be not in the same plane, the curve is said to be one of double curvature.
Formulae for the rectification of curves of double curvature are easily obtained. Thus, if the curve be referred to a system of rectangular axes in space, we shall have ds 2 =dx 2 + dy 2 + dz 2 .
Hence, if x be taken as the independent variable, we have
.-∕Zι + ⅛ + *‰, y ∖ dx i dx i J and similar formulae when either y or z is taken as the independent variable.
The equations of the curve are usually written in the form flx, y) = 0, φ(x, 2) = O; that is, the curve is determined by the intersection of two cylinders. The values of and deduced from these equations have to be substituted in the foregoing integral.
It is not difficult to determine a relation between the functions f and φ in order that the arc of the curve of intersection may admit of easy determination.
The simplest class is where (⅛Q —; for in this case
=∕*f 1 + ⅛ ∖ dx≈x+z+ const.
J ∖ fa)
For example, in the parabolic cylinder sj 2 = 2py + const., we have ^=-.
dx p
Accordingly, let ⅛ " ’
χ 3
and we get z= + const., hence the length of the curve of intersection of the cylindrical surfaces x 2 =2py + c, z 3 =6p⅜ + c ’ is immediately determined. In general, when y= ∖ f(f) is the equation of the first cylinder, and that of the second is represented by the equation z=⅛y*j f'(x) J- 2 dx + constant, the arc is determined by the above formula.
171. If we transform to polar coordinates by the relations sc=rcos Øsin φ, y=rsinβsin≠, 2=rcos≠, we get ds 2 = dr 2 + r 2 dφ 2 + r 2 sin 2 0dβ 3 ; hence, for the rectification of a curve of double curvature we have s=∕∕l + r 2 ^ + r 2 sin 2 ≠^ydr; J ∖ dr 2 dr 2 ∕ or s=∕*(r 3 + ⅛+r a sin 2 ≠^)⅛.
y ∖ dφ i uφ)
The latter gives for the length of the arc of a curve on a sphere, of radius a, 8'=a∕'(l + sin s ≠^)⅛.
If φ be const. = a, the curve lies on a right cone; and we have s=y(l + r 2 sin 2 <^) i dr. Cubature of Solids.
172. The method usually adopted, in seeking the volume of any solid, consists in supposing it divided by parallel planes into an indefinite number of thin slices. Then in finding the volume we may in the limit consider each slice as an infinitely thin cylindrical plate; and, consequently, represent its volume by the product of the area of the corresponding section into the indefinitely small distance between the parallel planes which bound it.
Thus, if the points in the body be referred to a system of rectangular axes of coordinates, and the system of parallel planes be perpendicular to the axis of x, then, representing the area of the section at the distance x from the origin by A x , the volume of the solid will be represented by fA z dx taken between proper limits.
Adopting a similar notation, the volume of a solid may be represented by
∕A j ,⅜, oγ∕Λ√Zz.
In the case of a surface of revolution, the sections are drawn perpendicular to the axis of revolution. Tims, if any curve, situated in the plane xy, turn round the axis of x, a plane perpendicular tβ the axis cuts the surface in a circle. The area of this circle is τy 2 ; consequently the volume between two sections, corresponding to the abscissae a and b, is represented by
∕* δ w∕ fdx. Ja
(1) Suppose the ellipse —- + = 1 to revolve round its axis of a⅛
Ct∕ 0^ then the entire volume of the generated solid is
< + X 1 - ⅜>- 2 '5f(1 - ⅞>-w∙
(2) If the parabola y==ax n revolve round the axis of x, the volume cut off by a plane at the distance X from the origin is f x ∙> o n , τrα 2 X 2n + 1 XY 2 y 0 2n + l 2n+l
(3) To find the volume of the ellipsoid
^. + Z + £„i a 2 + b 2 + c 3
Here the section at the distance z from the origin is the ellipse
≈ a , y s ~ 1 *L α 2 + b 2 - 1 c 3 ‘
The area of this section A s is
'(1 -⅞) αi ≈
accordingly the volume of the ellipsoid is represented by 2πabj~ ^dz≈⅛τtabc.
(4) To find the volume of the surface generated by the revolution of a cycloid round its base.
It is easily seen that the coordinates of any point on a cycloid, of radius a, are capable of being represented by z *=«(</> +sin φ), y=α(l + cos≠).
Hence the volume V generated is given by the equation
V = 2∙κa 2 i" (1 + cos φ) 2 dφ = Υ⅛πci λ ∕^ cos β ^⅛z^dφ — 5ιr 2 α 3 .
∙⅞ -×0 2
(5) To find the volume of the portion of the paraboloid x 2 u 2
-V÷A- = 22
l m cut off by a plane drawn perpendicular to the axis of z.
Here, the area of the section at the distance z from the origin is 2πz ∖ ∕l∕m. Hence, if c be the distance of the bounding plane,
V = 2πV⅛n∕ zdz≈πc 2l Hm.
* z 0
Consequently the volume is half that of the circumscribing cylinder.
173. Again, since any solid can be supposed divided into an indefinite number of elementary parallelepipeds, the volume enclosed within any boundary may be represented by JJJdxdydz , the limits being determined in each case by the nature of the problem.
For example, cubature of the ellipsoid
α 2 ∂ 2 + c 2 is equivalent to the determination of the triple integral fjfdιxdydz t for all values of x, y, z subject to the relation
4+⅛+4<ι.
Here, as in all other cases, we integrate, first, with respect to one of the variables, regarding the others as constant, and determine the limits from the given relation.
Thus, integrating with respect to z, and observing that the limiting values of z are ±c * /1 - ∙⅛ - ∙‰∙, we get
V Ct, 0 ' ’ -J∕ √ι-5-⅞ ife ⅛.
in which ar, y are connected by the relation
⅛ + Kl<l α 2 δ a ,
This integral is easily determined by making ∣-√Γ⅞≈w.
√rc 2
1 - — cos φ dφ t and V = 2δ^^l - -^ 5 -^cos 2 ≠d≠, where the limits for φ are and - — .
2 2
∕∙+ κ
But ∕ 2 cos i φdφ≈ ill-, J π 2
" 2~
hence V = ∙κbcj' - ~^dx = ⅛πahc i as before.
The geometrical interpretation of each step in the preceding demonstration can be readily supplied by the reader.
It may be observed that, in consequence of the symmetry of thθ ellipsoid, the preceding integrations might have been limited to positive values of x, y, z,— thus determining the eighth part of the entire volume. A similar remark applies to any symmetrical surface. It will also be observed that the determination of the volume of an ellipsoid is a simple case of the theorem given in § 157.
Similarly the volume included within the surface
(w<·
is reducible to the determination of the triple integral fff dxdydz extended to all positive values of x, y, z, subject to the condition (W≡ 1 ∙
Hence, by § 157, we get v.,-⅛Ha⅛).
r(ι + <±-±5)
Thus, for instance, the volume enclosed by the surface m≡F∙<
In like manner the volume enclosed within the surface w÷(÷y- ιfc ½∙∙ and so on.
174. From the preceding it will be apparent that every double integral may, in general, be represented by a volume.
As an example, let us consider the double integral z -2α J a ^' ia ∙ r - χ,t u ~Λ Jo ∕(,x,y)dxdy.
Here, since y>0 and <^-^f2ax-x 2 , and the limits of x are Cl
Ô and 2α, it is readily seen that the integral represents half the volume of the solid bounded by the surface z=f(x, y), by the plane of xy, and by the cylinder having as its base the ellipse
<z 2 ^ t ^ δ 2 a =
For instance, suppose the bounding surface to be the paraboloid 4 + i-2≈, l m then the volume in question becomes
* i - i ∕ Z (4÷⅛H
-⅛∕ a, ∣ ι< 2 ∞-^> ’ ÷ 3 ⅛(^-≈≈>≡ ∣*>∙
To integrate this, assume ∞=2α sin 2 0, and we get tt ≈ lβg - 3 j f 2 cos 2 0 sin 6 0dβ + ί 2 sin 4 β cos 4 βdβ
I Jo Jo r e ab [5a? 1 δ a ∖ = 8 ∖ l + rn / ‘
175. Again, the double integral
∕-X∕-Y
∕ ∕ Ax,y)dxdy, Jχ<Hy< i when the limits X, a⅛, Y, y 0 are constants, represents the volume bounded by the plane xy, the surface z=J(x, y), and the planes X =X, X = x 0 , y =Y, y —y 0 . Also, in the determination of this double integral the order of integration may, in general, be changed (§ 148); and this change in the order produces no alteration in the limits. The latter statement no longer holds when the limits of integration with respect to the first variable are functions of the second.
In this latter case it is of importance to be able to determine in each case what are the new limits when the order of integration is reversed. This can generally be best effected from geometrical considerations; thus, for instance, in the example of the preceding article, we readily find, when the order is reversed, the new limits of X to be a+a a ∕ 1 - ∣r a ∏d a ~ a a ∕I - ⅛-> and that the sub-
V b i z V br sequent limits for y are 0 and b.
As another example, let us consider the double integral
6
u≈f i a fχ,yjχdy. Jo Jo
If we take on the axis of ae a portion OA=α (fig. 10), and on ths axis of y, OB = b, and complete the rectangle OACB, it is plain from the equation that the point (x, y) is limited to the triangle OAC.
Accordingly, if the order of integration be reversed, we must suppose the area, instead of being divided into infinitesimal strips parallel to the axis of y, to be divided into strips parallel to the axis of X. Hence, the limits for x, when y is constant, are a and ψ; and the subsequent limits for y are b and 0.
Consequently, a -x rb a i ∕ a f fχ,yjχdy. Jo Jo ∙ χ o ja JL b
As an exemplification of the advantage of an interchange in the order of integration it Will suffice to take the double integral „ Γ a f x _ JTf)dxdV . JoJo ^∕(,a-x)(x-y)'
Here, interchanging the order, we have by the preceding v -ΓC fWy dx JoJv ^d(a-x)(x-y)'
But (§139), ∕ ∕∙∣√7p i i»r.
J y ∖ ∕{a-x ∖ x-f)
."/(Ο)} -
It may be observed that in many cases, when the order of integration is reversed, we get two or more double integrals instead of the original integral.
176. It is frequently found necessary to transform a double integral Jfflx, y')dxdy, referred to rectangular coordinates, to another referred to polar coordinates.
In this case, as in § 164, we substitute rdrdθ instead of dxdy, and the integral becomes JJ^fr cos 0, r sin fl)rdrdfl
The limits in the latter integral are determined from the equations which give the limits in the former.
For example, to find the volume comprised between the plane of xy, the hyperbolic paraboloid cz=xy, and the right cylinder (a5 -α)2 + (y -^≈⅛ 2 .
Here V— Jjjxydxdy, extended to all values of x, y, subject to the condition (x - a) 2 +(y - b) 2 <k 2 .
Assuming the origin of polar coordinates at the pointa, b, and transforming the equation, we get
1 ∕** ∕∙2n∙ τrabk 2 V = — ∕ ∕ (α + r cos 0)(δ + r sin fl)rdrdfl = , c ΛΛ c since.
∕-2π∙ ∕-2∏∙ ∕-2ιr
∕ sin0dfl=O,∕ cosfldfl-∙0, ∕ sin flcosfldfl=>0.
-×o -×o ∙Λ)
177. The triple integral ffffx, y, z)dxdydz can be transformed in like manner.
For, first, take x=p cos φ, y=-p sin φ, and the integral transforms into ffffp cos ≠, p sin φ, z)pdpdzdφ.
Again, assume z=r cos θ, p = r sin 0, and the multiple integral becomes JJJfr sin θ cos φ, r sin θ sin φ, r cos 0)r 2 sin fl drd∂dφ.
With respect to the limits in the new integral, it may be observed that, in this and all other cases, the new limits must be taken in such a manner that the transformed multiple integral shall comprise every element which enters into the original integral, and no more.
In particular the volume of any solid is represented by JJfr 2 sin fl drdθdφ, taken between limits determined by the boundary of the solid.
If this expression bo integrated with respect to r, we have
V=⅜j(7'∙ 3 s ™ 0 dθdφ, in which we must substitute for r its value determined by the equation of the bounding surface.
For example, let us investigate the volume within the surface (x 2 +y 2 +z 2 ) s = (α⅜ 2 + δ 3 y 2 + c 3 z 2 ) 2 .
Here we get r 3 =α 3 sin 2 fl cos 2 ≠ + ⅛ 3 sin 2 fl sin 2 ≠ + c 3 cos 2 fl, and, as the equation is symmetrical, we have fr π
V = ∣∕* 22 (d 3 sin 2 fl cos 2 ≠ + δ 3 sin 2 fl sin 2 ≠ + c 3 cos 2 fl) sin 0 dβdφ
≈⅛∕" 2 (2α 3 cos 2 ≠ + 2δ 3 sin 2 ≠ + c i }dφ = ~~- (α 3 + δ 3 + c 3 ).
√o 9
Again, the expression for the volume of the ellipsoid
^ + ⅜ + 7≡^ βl is represented by the integral π τr *J 2 Γ 2 sin fl dθdφ JO JO J sin 2 β cos 2 <∕> sin 2 fl sin 2 ≠ l cos 2 fl 1 ⅜
( ä 2 + & + c i I
Hence, since the volume of the ellipsoid is ⅜ιrαδc, we get
ιr π f 2 Γ 2 sin fl dθdφ __%αδc , Jo Jo I sin 2 fl cos⅛> sin 2 0 sin 2 ≠ cos 2 fl ∣^ ∣ 2 ’
i tf 2 + b 2 + c 2 i a result which admits of a direct demonstration.
178. The multiple integral u≈Jfff N dx 1 dxflχ. i dx i admits of a like transformation.
For, let % 1 =p 1 cos φ, sc 2 ≈p 1 sin φ, and it becomes ∕5∕V 1 p 1 dp 1 ⅜cfo 3 d¾, where V 1 represents the transformed value of V.
In like manner, if x 3 =p 2 cos ψ, x i ≈ρ 2 sin ψ, the transformed integral may be written ffffT 2 p 1 pgdp 1 dp 2 dφdψ.
Again, if p 1 = r cos fl, p. 2 ≈r sin fl, the integral assumes the form 3 r i sin fl cos fl drdθdφdφ, where V 3 represents the final form of V. In this case the values of ∞ 1 , x i , x 3 , x i , in terms of the new variables, are a⅛=r cos 0 cos φ, x 3 ≈r sin 0 cos ψ, x i =r cos fl sin φ, x i ∞r sin 0 sin ψ.
CvMdrature of Surfaces.
179. It is readily shown that the area of any cylindrical surface, bounded by two planes perpendicular to its axis, is equal to the rectangle under the height of the cylinder and the perimeter of its base; also that the surface of a truncated right cone is equal to the rectangle under its mean section and the length of the portion of any edge of the cone intercepted between the bounding sections.
In the evaluation of the superficial area of a solid of revolution, we proceed, as in § 172, by supposing the surface divided by planes perpendicular to the axis of revolution (fig. 11). Then the elementary portion of surface between two indefinitely near planes may be regarded as a portion of the surface of a right cone, generated by the revolution of the corresponding element of the curve round the axis. Hence, denoting the element PQ by ds, and PM by y, the area generated by PQ in a complete revolution round the axis of x is represented in the limit by 2πyds. Consequently, if S be the surface generated by the curve AB, we have
S = 2τfyds, taken between limits corresponding to the points A and B.
(1) Thus for the sphere, generated by the revolution of the circle x 2 +y 2 ≈a 2 round the axis of x, we have ds - i 1 + (≡)* ( * ife ^(1 +
Hence S = 2πfhdx=2πa(X-x 0 ), if X, x 0 be the limits for x.
Accordingly, the whole surface is 4ιrα 2 , i.c., four times the area of one of the great circles of the sphere. Also the surface bounded by any two parallel planes is equal to the corresponding surface cut out of the circumscribed cylinder, whose axis is perpendicular to the bounding planes.
(2) If the ellipse a ≈ a 4 .S' 2 βl revolve round the axis of x, we have
, Λ b i x 2 ∖ h, ' fe -(1 ⅛Jf)' fe > .∙. yds=~la" - c a -xfdx, where e is the eccentricity of the ellipse.
Hence, the whole surface of this ellipsoid is
4π∙- f (a 2 -e 2 x 2 ∕dx==2πb 2 + 2π^ 'si∏- 1 e. α√0 c
In like manner, if S be the surface generated by the revolution of the ellipse round its axis minor, we get
S = 2πj'xds =2ιr-p^(δ 4 ÷ a 2 e 2 y 2 fdy.
Consequently its entire surface is represented by o δ9 i ∕1 + A 2ιrα 2 + π— log -
e o ∖l-e∕
180. In connexion with surfaces of revolution, the following general propositions, usually called Guldin’s theorems, may be here stated.
(1) If a plane curve revolve round any external axis situated in its plane, the area of the surface generated in a complete revolution equals the product of the length of the generating curve into the path described by its centre of gravity.
(2) In like manner, the volume of the solid generated is equal to the product of the generating area into the path described by the centre of gravity of the area.
The former of these theorems is easily shown; for if y 1 be the distance of the centre of gravity of the curve from the axis of revolution, taken as that of x, we have y 1 s=fyds∙, 2πy 1 s≈2πJyds = S, which proves the theorem.
Next, if y i be the ordinate of the centre of gravity of the area A, we have Ay 3 = 2ydA = ffydxdy=⅛ ff i dx,
.∙. 2τry 2 A = ιr∕y 2 <fo!, whence the latter theorem follows.
181. In the general case of the determination of the quadrature of a surface we regard it as the limit of a number of indefinitely small elements, each of which is considered a portion of a plane that is ultimately a tangent plane to the surface. Now let dS denote such an element at any point of the surface, and dσ its projection on a fixed plane, which makes the angle θ with the tangent plane at the point, then we have d<r∙=dS cos θ, or dS=sec θ dσ.
Hence S =/"sec θd<τ, taken between proper limits.
If now the surface be referred to a rectangular system of coordinate axes, we may take dσ≈dxdy ∖ also, from an elementary theorem in surfaces, , , dz dz sec 0 = √l+p 2 + <p, where q °dy'
Hence we have s = j (7'(l +p 2 + q 2 ) i dxdy, in which the values of p and q are to be determined from the equation of the surface.
(1) For example, let it be proposed to find the portion of the surface of a sphere intercepted by a cone of the second order, whose vertex is on the surface of the sphere, and whose internal axis passes through the centre of the sphere.
Let 0 the vertex of the cone be taken as the origin (fig. 12), and the line joining it to the centre of the sphere as axis of z, then the equation of the sphere may be written x 2 + y 2 + z 2 ≈2az. Hence dz x dz y P≈~r a > g ≈; ax a—z dy a-z
∙ ’ ∙ √i+j> 2 +j 8 =-^-; consequently S>≈aj"J' — - , in which the limits are determined from the equation of the bounding cone. Let the equation of this cone be z 2 -A⅛ 2 + B 2 y 2 j then, eliminating z, the limiting values of x and y are connected by the equation
(1 + A 2 )z 2 +(1 + B 2 )y 2 - 2α√A⅛ 2 + B 2 y 2 .
Next, transform to polar coordinates by making x≈r cos e, y≈r sin θ, and we get
8-√7-ζ⅞⅛, √√ Va a -r i taken for all points within the curve r{(l + A 2 ) cos 2 β + (1 + B 2 ) sin 2 β} = 2α√A 2 cos 2 0 + B 2 sin 2 0.
Hence, since the curve is symmetrical, we get 1Γ a . f~T f v ∙ rdrdθ
S = 4α∕ ∕ . ,
√o √o √α 2 -r 2 where p _ 2αVA 2 cos 2 0 + B 2 sin a fl
1 + A 2 cos 2 0 + B 2 sin 2 0
∕* B τdτ ———- 2λ Again Jo √gτ⅝- a - '
S-Sα√^^ — 4 * αl
√0 (1 + A 2 ) cos 2 0 + ( 1 + B 2 ) sin 2 » √(1 + A 2 )(l + B 2 )
This result admits of a simple geometrical representation; for let D, E (fig. 12) be the points in which the edges of the cone lying in the planes y=0 and x=0 cut the surface of the sphere, and we plainly have r , r , 2α 2α
√1+A a, √1 + B 2
Consequently the area of the intercepted portion of the sphere is equal to that of the ellipse which has CD and CE as its semi-axes.
(2) If, instead of the cone, we had taken the paraboloid z=Ax 2 + By 2 , the area of the portion intercepted on the sphere is given, as in the preceding, by the equation ΊΓ
S - 4^∕* 2 (α - √α 2 ~≡K 2 )d0, where, from the equation of the bounding curve, we have p 2 2α(A cos 2 fl + B sin 2 ») -1 (A cos 2 0 + B sin 2 0) 2 l r „„ o , ∕*2 - dθ 2τra
27o A cos 2 0 + B sιn 2 0 √AB
This result admits of a geometrical interpretation similar to that in example (1).
Multiple Integrals.
182. The general form of a multiple integral may be represented by the expression z -x z -Y r U z -τ
∕ dx ∕ dy. . . ∕ du! dtf(x, y, . . . m, t),
√⅛⅛ ∙∕⅝ √<o in which f(x, y, . . . u, t) is supposed continuous for all systems of values of the independent variables x, y, . . . u, t included within the limits. Moreover the limits of each variable must be independent of the following variables, but may depend on the preceding variables.
In calculating the integral, the expression/(as, y, . . . u, t)dt is integrated between the limits T and t 0 , regarding x, y, . . . u as constants. Thus we obtain a function of x, y, . . . u. This function is integrated with respect to u between the limits U and w 0 , treating x, y . . . as constant. We thus obtain a function of x, y, ... independent of ∙u, t ; and so on for the subsequent integrations.
If the limits for each variable be constant, the integrations may be taken in any order, subject to such limitations as those given in § 148 for two variables. In the more general case, when the order of integration is altered it is necessary to determine, from the conditions of the problem, the new limiting values. This is usually a matter of much difficulty.
183. Continuing from § 178, the general problem of the transformation of a multiple integral by a change of variables may be stated as follows.
Suppose the multiple integral represented by Jf... , as 2 , · · · Xn)dx^ dx 3 ... dx„ , and it be proposed to transform it into another, depending on new variables u 1 , u 3 , ... u n , which are related with the original variables by a system of n given equations. This transformation implies three parts in general:—(1) the determination of ≠(¾ x 3 ,... x n ) in terms of u 1 , u 2 ,. .. u n ; (2) the determination of the new system of limits; (3) the finding the substitution for dx 1 dx 2 . . . dx n .
The solution of the first two questions is an algebraical problem, of which we have already considered one or two elementary cases. We now address ourselves to the third question, and write the integral in the form fdx 1 fdx 3 . . . f'dx n ^- i f'dx n φ{x 1 , x 3 . . . ).
In the integration with respect to x n , as stated in § 182, x 1 , x 3 , . . . x n -ι are regarded as constants. Accordingly, in order to replace x n by u n , it is sufficient to express x n in terms of u n , x 1 , dx x a , . . . x n -ι, and then to substitute yj^du n for dx n . Again, to transfonn the next integration, relative to dx n -χ, we suppose x n -ι expressed in terms of τι n - ∖ , un, x 1 , x 2 , . . . Xn-2, and we replace dx n -ι by d^ n -dun-i' By continuing this process the integral finally becomes of the form fff. ..fφ x ⅛⅛ . .. du x du 2 . . . du n , fff f ψl du n du n -ι du x where φ x represents the value of ≠(ar 1 , x 2 , . . . x n ) when transformed into a function of u 1 , w 2 , . . . w n .
Moreover, by § 102, the product dx n dx n -ι dx- i du n du n -ι ' ’ ' du 1 is, in this case, the Jacobian of the original system of variables x x , x 2 , . . . x n regarded as functions of the new variables.
Accordingly, for dx x dx 2 . . . dx n we substitute [table]
For instance, if JJNdxdy be transformed to new variables «, v, denoting by V 1 the value which V assumes, the double integral becomes JfT^uf.-f^dxdy, 1 . dx . dy , dx . dy where a/„= - τ -, f u =-f , al v =∙^-, fv≈-f · du a du dv i, dv
Again, if the coordinates of each point on a surface bo given in terms of two independent variables u, v, to find the transformed expression for the superficial area
Here dxdy becomes (rf u fv - dudv as before.
Also, since, from the equation to the surface, z may be regarded as a function of x and y, we have , , dz , , dz . ,dz ,dz dz = g f tt y ’ . - y ’ u √, dz = x' u z' r , - z' u x' v dx~x' u y ’ v - f u χl, ’ dy“x ’ κ f 1 - y' u x' v '
Accordingly the transformed expression is {(fuf v - √ u x ’.) 2 + (√ u √t, - tluf v ) 3 + (√ u a∕. - a∕ u √ r ) 2 } idudv.
For example, the coordinates of any point on the ellipsoid
α 2 ψ ft 2 ψ c 2 may bo represented by the equations x=a sinø cos≠, y=ft sin θ sinφ, z=c cosø; hence it can be shown that its total surface S is represented by S≈abcfΥ ∖ in θdθdφ j cos a 2 θ 1 ≡⅜ ≡ 2 ≠, sin 2 β cos 2 ≠) ⅜ .
√θ *' o ( c 2 ft 2 α 2 !
in which the integration with respect to θ can be immediately effected.
Again, the coordinates of any point on a sphere of radius a can be represented by the equations x=a sin 0√1 -⅛ 2 sin⅛>, y≈a sin ≠√1 -⅛' 2 sin 2 0, z=.a cos θ cos φ, where ½ 2 +X∕ 2 =l.
This is obvious, since the sum of the squares of these expressions s α 2 .
Accordingly
√ √ -√ √ a2 cos 0 cos cos 2 ≠ + F l cos 2 0)
β4 ^ φ °~ √l-⅞ 2 sinV√l-^sin⅞ ’ √ √ -z' √ _ a2 s ^ n0 cos 2 ≠ + A∕ 2 cos 2 0)
θ φ θ φ √Γ-⅛' 2 sin 2 0
2 > a √ - a ∕ √ g2 s ⅛ ≠(⅛ 2 cos 2 ⅜ + ⅞' 2 cos 2 0)
^ β φ θ φ ° √Γ^t 2 sιh 2 ≠
Hence we get
αW∞B⅝ + ⅜* c <> s ⅞)
VI -⅛ 2 sin 2 ≠ √1 -⅛' 2 sin 2 0
Consequently, since the entire surface of the sphere is 4πa 2 , we have
∕"^2^/Ύ (⅛ 2 cos 2 ≠ + ⅛' 2 cos 3 θ)dθdφ τr Jo Jo Vl-⅛ 2 sin 2 ≠ Vl -⅛' 2 sin 2 0 2
The well-known general formula of Legendre, connecting complete elliptic functions of the first and second species, follows at once from this last result.
184. In the case of three variables, adopting a similar notation, the integral fff ∖ dxdydz transforms into v>lfuf « - j∕u2 ’ ») + fwlfuXlg-d u s , ,) + α∕ w (y' u ≈ « — z ’ u fjfludvdw. For example, in the general transformation from rectangular to polar coordinates we find, as already observed, that 7∙ 2 sin θdrdθdφ is to be substituted for the element of volume dxdydz. This is but a particular case of the general transformation given in § 103.
The preceding formula of transformation for three variables was given by Euler in 1769, and afterwards generalized by Lagrange in 1773. Jacobi appears, however, to have been the first to have established the general transformation, in his memoir referred to in § 96. The method of proof here adopted is that given by Bertrand.
Ex. 1. In the case of linear transformations, viz., when x x ==a x u x + a 2 u 2 + . . . + a n u n , x 2 = b x u 1 + b 2 u 2 4- . . . + b n u n , x n ≈l x u x + l 2 u 2 + . . . +l n u n , we get dx x dx 2 . . . dx n =∆dιι x du 2 . . . du n , where [table]
Ex. 2. If ι r 1 =½, a ⅛=≡ 2 , α ,≡½, u x - w 2 u 3 the Jacobian is [table]
The value of this determinant is easily seen to be 4; hence fffydx x dx 2 dx 3 transforms into 4 fffN x du x du 2 du z . Ex. 3. As an additional example we shall take Jacobi’s method of establishing the fundamental formula of Eulerian integrals (§ 153).
Since
Γ(7) = ∕* e^¾ ’ ^ 1 dαr, Γ(τn) = ∕" e^vy m - χ dy do do we have
Γ(Z)Γ(τn) = ∕* f e~ x ~vχT' i y m -' i dxdy. do do
If now we transform by making x≈uυ, y≈u(l-v), the limits for u arc 0 and ∞, and those for υ are 0 and 1; also dv du du dv hence Γ(Z)Γ(τn) = i f e' ,i u z + m ^ 1 √ -1 (l -v) m - 1 dudv do do
= Γ(Z+τn)∕* v l ~ 1 (l -v) m - l dv, do . ∕* 1 l 1>- ∖m Γ(Z)Γ(Wl)
.∙. ∕ √- 1 ({ -τ) w - 1 dι>≈ ,-f j. do Γ(Z+wι)
185. In the more general case, where ic 1 , x 2 . . . x n are not given explicitly in terms of u x , u 2 . . . u n , but are" connected with them by n equations of the form
Fι(∙¾, ®2 · · · x∏, un u 2 · · · w»i) = 0 } F 2 (ar 1 , x 2 . . . a⅞,, u χι u 2 . . . u n )^0,
F„ -1(x 1 , x 3 ... x n , u x , u 2 . ..u n ) = 0, F n (a¾, x 2 ... x n , u x , u 2 ... u n ) = 0, we get, by § 99, [table]
where [table]
Accordingly the multiple integral ∣ JJ^ · ∙JTdxflx 2 . . . dx,, transforms into JJJ^· · . . . du n ∙
The limits in the transformed integral are determined by aid of the equations which give the limits in the original.
186. We conclude this short account of multiple integrals with a notice of the very general and remarkable theorems relative to integrals extended through a closed surface first given by Green {Essay on the Application of Mathematics to Electricity and Magnetism, Nottingham, 1828).
Let U, V denote two functions of the rectangular coordinates x, y, z which are finite, and have their first differential coefficients finite, for all points within a closed surface; then, since dv rτ d * y dx ∖ dx) dx dx dx i , we have g**÷>^*∙
the integrals being extended to all points within the surface.
Also, since the bounding surface is closed, any right line which meets the bounding surface, must cut it in an even number of points; hence the integral
- u > ¾)
where x, 2 , x 1 , U 2 , U 1 , &c., are the values of x, &c., for two corresponding points of intersection with the boundary by the infinitely thin cylinder standing on dydz, and by 2 is denoted the summation taken for all such values. Again, if dS 2 , dS 1 be the corresponding elements of surface, and α 2 , α 1 the angles which the exterior normal to the surface at each of these points makes with the positive direction of the axis of x, we have t⅞∕⅛=≡cos α 2 dS 2 = - cos a 1 dS 1 ; hence it is readily seen that the integral y⅛* s (v s ¾-u.⅛)
is equal to
^ r u ^cosαdS, taken for every element of the boundary, whether it consist of one closed surface or of several.
Accordingly, we get + ff^s dxd, ∙ ιds, -ff v ‰ 3 " ds ∙ in which the former integrals are taken for every point within any space, and the latter integral taken for each point on the boundary of that space. This may be written fff⅛ £ f * « --> 1 S
Taking the corresponding equations relative to y and z, we’ have by addition, fff ∖ dx dx dy -dy dz dz ∕ = cos a + -⅞^∙ cos β + cos γ)dS
J J ∖ dx dy dz ∕
∕7r τ √d 2 V d 2 V d 2 V∖ , . , fJj u ⅛ + ⅛ + ^) dxdydi -
Again, if dn be the element of the normal, measured outwards, at the element dS, we readily get dx „ dy dz cos a≈ -τ~, cos β = ÷ l , cos 7= ~ r -; dn dn ' dn dV dV n dN dV dx dy dz ’ dn
Hence Γff(dV dN dV dN dV dV ∖ . , . JJJ ⅛ ⅛ + -⅛ ⅜ + -di -d ι f 3d y∙ b
The latter equation is obtained by an interchange of U and V.
This is Green’s fundamental theorem, in the case where U and V are continuous functions.
187. The modification when one of the functions, U for example, becomes infinite for a point within the surface was also investigated by Green. Suppose this to happen at one point, P, only; moreover, infinitely near to P let U be sensibly =— , r being the distance from τ
P. Next suppose an indefinitely small sphere, of radius a, described with P as centre. Then it is clear that Green’s equation holds for all the space exterior to this sphere. Also, since
∕ cZ 2 d? d 2 ∖1
\cfø 2 + df i + dz i ) r = ’ the triple integrals may be extended throughout the entire space. Moreover, the part d ^ ue sur f ace °f the sphere is plainly infinitely small of the order of a. It only remains to consider the value of JJ ∖ r dS taken over the surface of the sphere.
But, since - 7 -= - 2_ this becomes - 4τrV 1 , where V 1 is the value dn a l of V at the point P.
Hence, denoting d 2 . d * , cp d^ + df + d^ γ 7i, we have ilxdydzVV·H -JJV ~ dS
a JJfdxdydzXvJJ ~jyT dS + 4πV 1 , where, as before, the double integrals are extended over the bounding surface or surfaces, and the triple integrals taken throughout the entire space enclosed.
These theorems of Green have been generalized by Sir W. Thomson; thus, if α be another continuous function of x, y, z, we get, by a similar treatment, instead of Green’s first equations, di 9 dV ∖ ) , , , + -^√ α ~dz) ( dxdydz
÷⅛(∙ ’ ⅞)!"^∙
with a corresponding modification when one of the functions becomes infinite at one or more interior points.
In the case of many-valued functions, another modification of Green’s theorem was established by Helmholtz (“Geber Integrale der Hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen,” Grelle, 1858).
Elliptic Integrals.
188. Attention has hitherto been restricted to integrations of rational algebraic functions, of logarithmic or circular functions, or of such functions as could be transformed to depend on these; or, if irrationalities were introduced, they were such as involved the variable under the radical in no higher than the second degree. But the founders of the infinitesimal calculus early perceived that many integrals did not admit of expression by means of these elementary functions with which they were familiar. Apparently it was the geometrical interest attached to such integrations which first attracted notice. Thus James Bernoulli published, in the Acta Eruditorum for 1691, a paper on the hélicoïdal parabola, in which we meet with the idea of comparing arcs of one and the same curve, which cannot be superposed.
This spiral is the locus of the extremities of the ordinates of a parabola when its axis is rolled as a tangent to a fixed circle, the ordinates being measured towards the centre. The polar equation of th e locus is (rt-r) 2 =2α⅞w (see fig. 13). Hence the — 7*)^
1 — » so that, if s 1 be the arc contained be tween the values ⅜α and ⅜α + c (=AG) of r, and s 2 the arc between the values ⅛α - c ( = AN) and ⅜α, we have
∕*2 +e ∕, r 2 (a-r) 2 , f~t f~ r 2 (α-r) 2 51 ~j i V 1 + ^Wt⅛ 2 dr, 9 ' i j*- c 'v 1 + ~α¾ 2 dr '
2 2
Now, putting in the former r≈⅛a+z, and in the latter r=⅛α-z, we find _________
aι ~ 8t z∕> ∙v^ 1+ ^G 2- t) dz ' , whence we conclude, as Bernoulli did, that even in curves whose rectification has not yet been effected, parts may be assigned which are equal though dissimilar; such as BG=AN, and GI = NI, where AI = ⅛m
John Bernoulli, following up this discovery of his brother, proposed to find for a given curve another, such that the sum or the difference of two arcs, one on each curve, may be expressed by arcs of circles. In the case of the cubical parabola he noticed that the two curves reduce to onè curve, in which, without effecting the rectification, pairs of arcs could be found whose difference is rectifiable.
189. The Count Fagnani next proposed in 1714 the problem, “given a portion of the parabola whose equation is x i =y, to find another portion of it such that the difference of these two parts may be rectifiable.” In the following year, not having received a solution, Fagnani published his own, with greater generality, as follows.
rn÷2
O ae 2 Taking a constant, and m any real number, let y=—— —bo β j m + 2 ~ a a the equation of a parabolic .curve (compare Ex. 2, § 166). Let t be the portion of the tangent between any point (xy) of the curve and the axis of abscissae.
Then by the equation of the curve, • i ≈- .∕ι + (- ∖ m m + 2 V + ∖α) ’
and the arc of the curve s W ι+ {τY dχ ∙
Integration by parts gives (see fig. 14)
^£7qTy· = ”' rr ·- (,,A - P1!) · '
if P, P 1 have the abscissae x 0 , x 1 ; also m ∕∙2 ’ ι r∕v ~Ι' zgv √≈arcQQ 1 -(Q 1 S 1 -QS), t ° √ 1 ÷(α)
if Q, Q1 have z 0 , z i , as absciss».
Now if the former integral can be transformed by introducing a variable z so that the function under the sign of integration may remain unaltered, and thus the former integral may pass into the latter, we may equate arc PP 1 - (P 1 R 1 - PR) = arc QQ 1 - (Q 1 S 1 - QS). Thus the question is to determine an integral of the differential equation dx ' dz ;?wT:
√ ι WV>√ιy ’ or of 7~—; + 7 z r r tfe = 0 .
√ 1 ÷(⅞)" √>√÷Γ
Fagnani gave the following solutions. For m = 4 the curve is the cubical parabola, and the relation between x and z is xz=a 2 . For »1=3 the equation is satisfied by the relation
MM)-*·
For τn = 6 the curve is as in the problem proposed, and the relation is ω i b(⅜y-d=(⅛y-'∙
190. Passing over Fagnani’s investigations relative to the lemniscate,—such as his discoveries of the method of doubling or halving any arc, of dividing the quadrant into equal parts in number 2.2 m , 3.2 m , or 5.2 m , discoveries which prompted to their author the wish, since executed, that on his tomb a lemniseate should be inscribed in memoriam, — we must mention his well-known geometrical theorem, that on the circumference of an ellipse, in innumerable ways, pairs of arcs can be determined having their difference expressible by a right line (first published 1716). His method is as follows. If we have h, l, f, g constants, and {fhx 2 z 2 ) ’ + (flx 2 )* + (∕Zz 2 ) ’ + (gZ)∙=O, then, first, the sum ∕ dx ∖ ∕hx 2 + l Γdz ∖ ∕hz 2 + l hxz .. 1 . - , n . f ffx 2 +g J ∖ ∕fz 2 + g -fl
, ,, ,. 1 ≈V -h , and, secondly, the same sum = a + — — when s= -1.
In the former case the relation gives g ∖ ∕-flx 2 -gl ∖ lfhx 2 +fl
Introducing this into the first integral, and the corresponding value of x into the second, the sum of the integrals becomes Γdx ∖∣ -I Γdz ∖ l - l J z ∖ tf J x ∖ ∕f
But differentiating the relation, and dividing by 2fxz, we find hzdx+hxdz + l- + l-=0∙, z x whence, substituting, the sum is found to be as stated. A like treatment yields the formula when s=-l. This theorem is applied in its former part to elliptic arcs. Let us call the axis major 2a, the parameter p, and the abscissa x ; then, if h=∙p-2a, the element of the arc AB (fig. 15) corresponding to the abscissa CD = x can be shown to be cføV hx 2 + 2a i ’ f2a 3 -2ax t
If now Z=2α 5 , ∕= - 2a, and g≈2a 3 , this becomes the former differential in (1); and it appears that, taking another abscissa
√A≈ 2 + 2α 3 we have . _ , _ hxz ι τr arc AB + arc AF = -+ K.
To determine the value of the constant K, letx=0; then AF becomes the entire arc AG, hence . τ , hxz arc AB - arc GF = - .
The second part of the theorem is applied to the hyperbola (fig. 16). Calling HA = 2α, the parameter p, and x the variable abscissa CD, and putting h=ρ + 2a, the element of the arc AB is easily found to be expressed by dx^hx 2 -2a 3 t √2ακε ’- 2α 3 *
whence, identifying, we have Z=° - 2α 3 , ∕ =2α, gr=-2α 3 , and,
∩ ~~ it ·
assuming another abscissa CE = ^= , —the theorem gives √Λic 2 -Λα 2 arc AB + arc AF = -⅛J + K .
αv 2α
Taking another pair of abscissae t, u, similarly related, we have . _ . , tu f fh τr arc Ab + arc A∕= + K , and by subtraction the arbitrary K is eliminated.
191. In order to be able to state the results more concisely, it is desirable here to explain in anticipation the notation introduced by Legendre, which has since generally prevailed.
If the position of a point on an ellipse be expressed by the coordinates ir=αsin<∕>, y = bcosφ, it can easily be found that, denoting the eccentricity of the ellipse by κ, the arc reckoned from the extremity of the axis minor A to the point B determined by φ is expressed by arc AB r⅛ , τ —=— δ -,
— =∕ V1 - κ-sιn 2 φdφ. a Jo
This Legendre writes arc AB .
___ = W
If it were desired to indicate also the quantity κ, which is called the modulus of this elliptic integral, he writes it E(κ, ≠), and calls this an elliptic integral of the second kind, for reasons which will soon appear. The quantity φ is called the amplitude of the elliptic integral, and its geometrical meaning is the eccentric angle measured from the axis minor of the point for which E(κ, ≠) measures the arc. For brevity also he adopted the notation √l-κ 2 sin 2 ≠ = ∆(κ, ≠), or = Δ(≠) when it is unnecessary to mention κ.
192. Legendre, proceeding to rectify the hyperbola = 1, first assumes x≈asecθ, and this gives the element of the arc = cos <j0Vβ 2 + α 2 sin 2 0; but to have a radical similar to that of the arc of the ellipse he had recourse to another notation. Determining φ by the equation
β≡ 1 , „ , α√ 1 — κ 2 sin 2 √>
«·= — tan φ, where c 2 =α 2 + β 2 , we get x= - — » a c ’ & cos φ in which a = cκ. The arc of the hyperbola is in this way found to be g 2 B dφ c J cos 2 φ∆φ ’
Again d(tan≠∆≠) = — κ + ∆φdφ, wnere κ 2 + κ' 2 =l .
cos i φ∆φ ∆φ
Hence the hyperbolic arc AB of which the extreme ordinate is BD, or cκ' 2 tan φ, is AB *=c∆φ tan φ - c /' ∆φdφ + cκ' 2 f ⅛ φ - . Jo Jo λ Φ
The geometrical meaning of φ is easily determined by taking the circle on the transverse axis, and joining its intersection with the tangent at B to the centre; φ is the angle the joining line makes with the perpendicular on the tangent. It is seen thus that the arc of a hyperbola depends, not only on the integral which gives the arc of an ellipse, but also on ∕* , which is called an elliptic in-
Jo δ Φ tegral of the first kind, and denoted by F(κ, <∕>), φ and κ being called amplitude and modulus as before.
193. When this is applied to the formulae of Fagnani they become for the ellipse, calling ψ the value which φ has for the point F, sin ψ = S2≡-≠ , arc AB - arc GF = aκ 2 sin φ sin ψ = a κ 2 sin ≠ cos φ ; γ ∆φ ∆φ and it is easily found that the expression on the right is the length on the tangent at either B or F intercepted between the curv*e and the foot of the central perpendicular let fall upon it.
In application to the hyperbola, similarly, sin ψ= c θ s J 6 , and arc AB + arc AF=- r — 1- const.
∆φ sin φ cos φ
Now the length of the tangent between the point of contact φ and the foot of the central perpendicular upon it is c tan φ∆φ ; hence denoting by E and F the complete functions E(κ, ⅛π), F(κ, ⅜π), the value of the constant can be determined;
. arc AB÷arc AF ∆φ ι e . c sin φ cos φ ’ and this value of the constant is the difference between the entire hyperbolic quadrant and the length of the corresponding asymptote.
194. Landen, continuing these investigations in 1775, considers the hyperbola whose semi-axes are a≈m -n and b = 2 ’ ∖ ∕mn ; then, writing CP= {m-n -i 2 } i (fig. 17), he notices that DP-AD=∕i^Ξ⅛± 2 l⅛.
J ^ m- ∖ -n -t 2 ’
Then he says it is well known that, in the 'ellipse whose semi-axes are m, n, the arc z from the conjugate axis to the point whose abscissa is x is
√(⅛⅛) l < fa ∙
∙, . ml — Ti 2 where g is —; m 2 and therefore in the ellipse whose semi-axes are m + n and e BJmn, for an ab- scissa = ?7Γ7ι t, the arc = ί i )⅛. But further, in the Hl> — H ∕ ∖ * ,n ∕ * z ×m-n ~t 2 J ellipse m, n, the length t of the tangent at the point whose abscissa is x, to the foot of the central perpendicular on it, is . (m 2 -x i ∖ h
; whence 2gx 2 =gm 2 + Z 2 - √(wι 2 - u 2 ) 2 - 2(w 2 + n 2 )i 8 + i 4 . Differentiating this, since
≈√⅛⅛, .∙. 2≈-. i + i∕(⅛^∖⅛ +i ∕∕'⅛ll s ‰
J t J ∖ m-n - t 2 ) J ∖ mj c n -11) which is a relation between the hyperbolic arc, the two elliptic arcs, and a portion of a right line. Landen remarks on his discovery, “Thus beyond my expectation I find that the hyperbola may in general be rectified by means of two ellipses,”—a result apparently of secondary importance compared with the method it is attained by, which involves the principle of what after Legendre is known as Landen’s transformation.
195. But, simultaneously with the geometrical interest thus developed in these integrals, they also attracted attention to their more algebraic relationships. James Bernoulli (1694) devoted particular attention to the “elastic curve,” which is defined by the equation dx≈± - J- , with a view to construct it by the quad-
Vα 4 - y 1 rature or rectification of a conic. Maclaurin (1742) gave such a construction, depending on the rectification of the equilateral hyperbola; and in like manner constructed by the aid of arcs of conics the integrals of expressions such as dx dx
√α√l+x 3 ’ (lψx 2 ) i ’
and others like them, which can be reduced to elliptic differentials. D’Alembert (1746) extended these results. His paper in the History of the Berlin Academy treats a number of differentials, whose integrals are of the same forms as those by which the arc of an ellipse or hyperbola is expressed. When a differential expression can be reduced to the differential element of the arc of one of these curves, D’Alembert calls this the integration of it by means of an ellipse or hyperbola. This paper is of a purely analytic character, and is penetrated by a tendency to classification of elliptic differentials similar to that which has effected in the works of Legendre such important services in the development of analysis.
196. In the works of Euler, of which from 350 to 400 quarto pages are concerned with this department of our subject, the geometrical and analytical aspects alternate. His first investigations are in the St Petersburg Commentaries (1761), on the integration of the differential equation mdx __ud±_ .
Vl - se 4 Vl - τ∕ 4 and it is remarked that the differential equation of a more general form mdx ndy
√A + 2Ba3 2 + Cz 4= √A + 2By 2 + Cy 4 can be completely integrated by an algebraic equation, provided the numbers m and n are rational. He extends the same method of integrating to the apparently more general equation mdx ndy
√A + 2Bz + Caj 2 + 2D≈ 3 +Ex 4 √A + 2By + Cy 2 + 2Dy 3 + Ey 4
In the next paper in the same volume Euler determines on the quadrant of an ellipse two arcs whose sum can be expressed geometrically, reproducing many of Fagnani’s formulae.
197. Euler’s most important investigations are collected in his Institutiones Calc. Ink , vol. i. sec. 2, cap. vi. His method, being essentially as follows, has much analogy with Fagnani’s, given in § 190.
Let the equation (ax 2 + 2a'x + α")y 2 + 2(‰c 2 + 2b'x + b")y + cx 2 + 2clx+d'=l>y 2 + 2^Λ.y + 'S = 0 (1), or (ay 2 + 2by+c)x i + 2(dy 2 + 2b'y+c f )x + a"y 2 + 2b"y + <l ’ =T ’ x 2 + 2(ix + 'R=0 (2), subsist between x and y. Differentiating either, we get
(Px+Q)cte + (Ly + M>⅞∕=0 (3).
Further, by (1, 2)
(Pz + Q) 2 = Q 2 -PR, (Ly + M) s = M 2 -LN, . . . (4); taking the roots positive, (3) becomes
** c + > dy - ≈0 (5),
√M 2 -LN √Q 2 -PR
in which the radicals are respectively functions of x and of y expressible from (1) and (2).
If the functions under the radicals are to be severally the same functions of x and y, the following conditions result :—
δ 2 - ac = α' a - αα", 2bb'-ad-alc≈ 2α , δ' - αδ" - al'b, bb" - a ’ d, 2b'b" - a'c" - a"d = 2b'd - δc" - b"c, b" 2 - al'cl' =c' 2 - cc".
The values of c and c' being substituted from the first and third equations, in the second we get a!=b. Whence first and third give α''≈=c, δ" = c'; and the others are identical. (1) thus becomes ax 2 y 2 + 2bxy{x + y} + c(x a + y 2 ) + l·b'xy + 2c'(x+ y) + c' = 0 . (6).
Hence (5) takes the form + dy 0 √A+2Bx+Cx 2 + 2Dz 3 + Ea3 4 √A + 2By+Cy a + 2Dy 8 + Ey 4 ’
or more briefly dχ , dy a
√x + √γ ’
of which the obvious transcendental integral is ∕ dx Γ dy
√xt∕ √γ" const ∙
where A = c' 2 -cc", B = 2δ'c' - be" - cc',
C = 4δ' 2 -αc"-c a -2δc', D - 2bb' - ac - be, E = δ 2 -αc.
Further, from (4)
√Y = (αy 2 + 2by + c)x+ by 2 + 2b'y+cl, ^fX≈(ax 2 + 2bx+d)y + bx 2 + 2b'x + d. ∏ ence ^~J-^axy + b{x + y} + 2b'-c.
Squaring, we get, by (6), &c., the algebraic integral
∕√x-√γ∖ 2 ∖ x-y ) = E (* + 3∕) 2 + 2d (= c + y) + (2* ’- c) 2 - αc".
The constant on the right, involving an arbitrary quantity, if A, B, C, D, E are known, may be taken as the constant of integration.
198. In Euler’s first paper in vol. vii. of the St Petersburg Commentaries, he uses the ‘ ⅛quatio canonica’ ’ 0 = α + -y(iκ a + y 2 ) + 2bxy + ex 2 y 2 as a starting point for establishing his theorem of addition, and, introducing the notation
∕* 2r A'+ B⅛ a + C'a· 4 n(ae) = ∕ T— — =dx , k ' √ 0 VA + Cae a + Eae 4 establishes the equation
∏(x) + ∏(y) - ∏(a) = Γ _ β' _ σ + ⅛ 2 Εσ “I
√A L 2 6A J
»h«™ a (a + c y s + E y 4 )+ a (a + caes + e ^)
A-Eae a ÿ 2
Now, when B'=0, C'=0, these equations represent the theorem of addition for Legendre’s first kind of elliptic integrals, and when a = 1 1 C= - (1 + κ 2 ), E = κ 2 , A' = l, B'= - κ 2 , C'=0, they become the theorem of addition for his seoond kind. Thus it appears that already in 1761 Euler was acquainted with this fundamental theorem, of which he gave many applications to the comparison of elliptic arcs.
199. But it seems to have been Euler’s paper ‘ ‘ De reductione formularum integralium ad rectificationem ellipsis ac hyperbolae” (Novi Comment,., x. p.3-50, St Petersburg, 1766), which impelled Legendre to his investigations. With special cases of the general relation, § 197 (1), betwee n x and y, Euler transforms integrals contained in the form ∕ f+ y f dx, and distinguishes whether the integral has one
V tC - f^
of the five significations, arc of ellipse, arc of hyperbola, or either or both of these along with an algebraic part,—collecting in one general investigation the results of Maclaurin and D’Alembert on the rectification of conics. Here we find the words,—regarding the desirability of a suitable notation by which elliptic arcs may be as conveniently expressed in calculation as logarithms and circular arcs are at present, “such signs,” he says, “will afford anew sort of calculus, of which I have here attempted the exposition of the first elements,”—which Legendre cites in the preface to his great work in 1825 as having remained unfulfilled but for his own labours continued till that date from his first publications on the subject in 1786.
200. In vol. iv. of the Miscellanea Taurinensia, Lagrange deals with the integration which had been given by Euler, remarking that it was due only to a kind of lucky accident. This indeed Euler himself admitted when he stated that he had not obtained this result by a regular method, but “potius tentando, vel divinando,” and recommended mathematicians to seek a direct method. Lagrange here lays down the principle that, when the integral of a differential equation of the first degree cannot be found, the equation should be differentiated; and, combining the result with the given equation, an integral equation of the first degree different from the proposed may be found. Then by means of these two the first differentials may be eliminated, and the result is the required integral. If this fail we may differentiate once more, and try to get a new equation of the second order, and so on. This enabled him to give a deduction of Euler’s equation, which Euler received with the greatest admiration, and gives nearly as follows in his Institut. Calc. Ink, iv. p. 466.
Writing for brevity
A + Bz + Cz 2 + Dic 3 + Ez 4 =X ) , 1 .
A + B 2 ∕ + C 2 ∕ 2 + Dy 3 + E 2 ∕ 4 =Y ∣ * · ‘ ' ' w suppose the differential equation between x and y to be
⅛ + ⅛- 0 < 2 >-
Regard x and y as functions of a variable t, and replace (2) by the following
⅛.√χ,⅛.-√γ (≡)∙
Assuming x + y=p, x-y≈q (4), weget ⅞=√χ-√γ. ⅛=√X+√Y, (β )> (Lt at at i 2 which last is, by (1),
⅛ = B + C(z + y) + ⅜D(x a + y 2 ) + 2E(z 3 + y 3 ). . . (6). at
We also get, from (5), ⅛ -⅛L = X-γ at at = (,x-y') {B + C(x+y) + D(iB 2 + ≈y + y 2 ) + E(a! + ι∕)(a5 2 + y 2 )}, which, combined with (6), gives, by (4), cPp dp dq ,, ιτ , , . . 2 dfl dp ∖
'⅛~Λ ⅛-ΛiD+⅛). ··». 7 s ∙(7 U ' ’ ' Multiplying by ⅛ this gives on integration, with F a constant,
(⅛ g) ’-⅛+⅛∙+Fi whence, replacing values,
∕*∕γ r — ∖∕v ∖ 2
(7 , „ ) =D(i r + y) + E(x + 2 ∕) 2 + F, ∖ x - y ∕ which is the same result as Euler’s (§ 197).
A principal advantage of this method consists in its admitting of generalization, which Euler’s method, depending on the solution of a quadratic equation, excludes. But Lagrange fails to apply it to the case of X and Y being arbitrary polynomials respectively in x and y ; all assumptions lead back to the forms of X and Y in (1).
201. In a paper of Euler’s in the St Petersburg Transactions for 1771, an angle is introduced as the variable into the integral for the arc of an ellipse. In another paper, in the Novi Commentarii, 1767, he had also remarked that, in a differential such as dx
VÂ+Bae+ Car 2 + Dx 3 + E≈ 4 ’
by a substitution of the form ¾=- 2 + f ι the odd powers of z under j nz + b r the radical can be abolished; and by a like substitution, removing the odd powers of y, Euler treats, without any loss of generality, the differential equation in the form dx dy x/Α + Cae 2 + Eae 4 x/Α + C ?/- + 1⅛ 4 This furnishes an essential simplification of the process of calculation, and leads to the results of his important and remarkable “Plenior Explicatio” in the St Petersburg Transactions for 1781, This contains in fact a proposition which includes the theorem of addition for all three kinds of elliptic integrals of Legendre,
202. Putting ∏(z)=∕* ... '⅛∖=≈.,
√o √l+≡ 2 + n≡ 4 where Z is an even function of z, ∏(x) + ∏(y) - ∏(z) can be exhibited as an algebraic function of z, y, and z, provided a certain algebraic relation holds between x, y, andz. “But now,” observes Euler, “I have noticed that the same comparisons may be instituted if for Z be assumed any rational function of z 2 , as suppose one of the form
F + Gz a + H⅞ 4 + Iz a + Kz 8 +.
f+ gz 2 + Λz 4 + iz i +k^ + ’ in this case, however, the difference between the sum of two such formulae and a third is no longer found to be an algebraic quantity, but can always be expressed by logarithms and circular arcs, so that the investigation is much more extensive than I hitherto conceived.”
To establish this, assume z 2 + y 2 - z 2 + 2xy^f 1 + wιz 2 + πz 4 - nx 2 y 2 z 2 =- 0 . . . (1), or, writing Vl + wιz 2 + nz 4 =∆ (2), z 2 + y 2 - z 2 + 2xy∆ - nx 2 y 2 z 2 = 0 (3).
Solving in turn for x and y, this gives z(l - ny 2 z 2 ) + y∆ = zV 1 + my 2 + ny i ∣ ,, ·.
3∕(1 - nx 2 z 2 )+x∆ = z√1 + rnx 2 + nz 4 1
But, by differentiation, we know that d¾ + dy θ
√1 + τnz 2 + 7iz 4 √1 + my 2 + nf ’ ’ ’ which may be written thus d χ dy t e . ( 6 ) - y(l-nx a z 2 )+x∆ x(l -ny a sp)+y∆ *
For x≈0, the relation (3) gives y=z ; whence (5) integrated gives the known theorem of addition for integrals of the first kind.
Now let X,Y be the same functions of x 2 , y 2 as Z is of z 2 ; then ... ^X,dx Υdy writing 1—, j = dV . .... (7), ∖ ll+mx 2 + nx i ∖ Zl + my 2 +nf the quantity V may be found as follows.
We can eliminate by (5) either dx or dy ιrom (7); but, as there is no reason to consider V as a function of x or of y specially, introduce a new independent variable u=xy. Then we may replace (6) by dx=[y(l-nx 2 z 2 )+x∆]sdu, dy=-[x(l-ny 2 z 2 ) + y∆]sdu . . (8),
, . 1 1
where we take s=- 5 ≡ · y 2 -χ- dx dy zdu
Vl +mx 2 +nx i ^vl + my 2 + ny 4 y i ~ χ which give dV= (9).
V — X
But now s is a function of xy and z 2 + y 2 , or of u, since by (3)
y*2 —· ⅛cθ
z 2 '+ y 2 ≈z 2 - 2u∆ + nu 2 z 2 .
Y — X
Thus, putting z- s 2 =U, wehavedV = -Vdu. Substituting this y ~ χ in (7), and integrating, we have Γ u
∏(z) + ∏(ι∕) - ∏(z) = - Vdu,
-Λ)
since, for z=0, y=z and u=0.
As an example, . f „ a + bz a ,·. Y-X alb-all ~ a! + b'z 2, 1 ° n y 2 -x 2 α' 2 + a'b'(x 2 +y 2 ) + 5'⅛ 2 y 2 ’
,. . . ττ (α'⅞- ab')z a ls, - α' a + a'b'z 2 + 2a!b'∆u + (bl^ 2 + α'δ'nz 2 )u 2 ’
Thus, if we take a+bz a dz
~ Λ a ' + b ' z * √l+rn 8 +nz 4 ’ we have in which after integration u is to be put *=xy∙ By assigning special values to the constants this equation can be applied to each kind of elliptic integral.
203. In the Théorie des Ponctions Analytiques of Lagrange there is found the remark on the relation between the summation of elliptic integrals and spherical triangles which is involved in the formulae of § 200, and. of which we have given the essential formulae in §§ 34-36. But Lagrange’s paper “On a new method of the Integral Calculus for Differentials affected with the square root of a polynomial of not more than the fourth degree” (Mémoire de VAcad. des sciences, 1784-5, 2d part, Turin, 1786), contains additions of much greater importance to our subject.
Taking p' ≈p + f fp 2 -q 2 , q , -=p-Vp 2 -g ,a , ρ'' =y + √ p' 2 - g ' 2 , g " =p' - ∖ ∕p' a -<ρ, ρ"' =ρ" + √p" 2 - 3 " 2 , q'" =ρ" - ^ρ" 2 -q" 2 .
Further, writing for brevity,
R = √(l±i⅜ 2 )(l±<z¾ 2 ), R ’ =√(i±mW). R"=√(i±pV 2 )(i±√V 2 ); • •••••••••· and √=-^- √>- y™ ι p'- w ... y l±q 2 f , u l±f 2 f 2 ' U 1±∕V 2 we have the equations 2 ⅛⅜ i V 2 -l + R z y ±2ρ 2 ’
, a , ⅛ g V 2 -l + R" y ±2p z2 ’
±A" 1 -ι+r.
v ±2ρ" 2 ’
⅛=F⅛'∙∙∙' '
Now, assuming q<ρ, we have ρ<lf<ρ" ∖ and, as q r =q. — , Pl q>f>(f, &c., without limit.
Lagrange gives another system of equations of reduction following from the above series when continued backwards, so that the series are • · · P» JPi» P, P', P u · · ∙
···?«, 9 ’ > ·
The equations P =Pι + √Pι 2 ~g ’ ι a , S=Pι~ √JPι 2 ~gι 8 , Pι=P∙2 + ∖ fPz~T2, gι=P 2 -√Λ 2 -g 3 a , give conversely
2h = ⅛0 j +g), gι= ’ √2^>
J⅛=⅛(Pι + Si), g 2 =√2h7ι, and the terms p, p l , p. 2 . . . decrease, while q, q 1 , q 2 . . . increase. Now, putting j ι±gι 2 yι 2 ' λ iiqj 2 y-2 i, '"
Rι=√(⅛ 2 O⅛ s y 1 2 ), b 2 =√(i±Λ ’ W)(i÷g 2 2 y 2 2 ),. · we get
,. 2 -. ⅛V- 1 ÷ r &c antl 2/1 ±2 i > 1 ≡ ’ ^ c ∙ , and R - r;~ r 2 ~ &c ·
The former series are essentially the transformation known by Landen’s name; the latter, indirectly, are (§ 214) the transformation which Gauss published in 1818, and. on which his theory of the arithmetico-geometric mean is based.
204. We have thus glanced at the most important contributions to this branch of our subject previous to Legendre. His “Mémoire sur les intégrations par d’arcs d’ellipse” (Histoire de l’Acad., 1786) appeared a few years after Euler’s death (1783). The geometric basis is here almost abandoned. Establishing with ease and elegance the theorems of Fagnani, Landen, and Euler, there are perceptible traces of a coming theory of transformation in the analytical conception of these theorems. Legendre’s Mémoire sur les Transcendantes Elliptiques (Paris, 1793) contains the division of elliptic integrals into their different kinds, the reduction of integrals of each kind to the simplest normal forms, and the calculation of elliptic integrals by most accurate modes of approximation. All these investigations are collected in Legendre’s Exercices (Paris, 1811-19), and later in his Traité des Fonctions elliptiques et des Intégrales Eulériennes (Paris, 1825-6, 2 vols., suppl. vol., 1828).
205. “It is Legendre’s undying glory,” said Lejeune Dirichlet of this great work, “to have recognized in the discoveries we have just mentioned (of Fagnani, Euler, Landen, and Lagrange) the budding of a mighty branch of analysis, and by the toil of half a life to have erected on these bases an independent theory which embraces all integrals containing no other irrationality but a square root under which the variable rises only to the fourth degree. Euler had already noticed with what modifications his theorem can be extended to such integrals; Legendre, starting from the happy thought of reducing all these integrals to fixed canonical forms, attained the knowledge, so important for the development of the theory, that they group into three essentially different kinds. Submitting then each kind to a careful investigation, he discovered many of their most important properties, of which chiefly those which belong to the third kind were very obscure and inaccessible. Only for the most persistent tenacity, which ever anew led the great-mathematician to his subject, did the victory at last declare itself over difficulties apparently insurmountable by the weapons at his disposal.”
206. Having shown that the integral, where P is a rational function of se, and R∙=(α+0a! + γaj 2 + δ≈ 3 + eΛ! 4 )*, can bo reduced to the fixed fundamental forms ∕^4r > ∕^~⅛~ > ∕*~vr-
J R J R J R
— dx ', , Legendre removes, by aid of the linear transforma- (l + τιaj)R
tion x = } the odd powers of the variable from the poly-
nomial R 2 , and shows, by enumeration of cases, that —- can always
XV
be reduced to the form md ^ ’ — where c is a quantity less Vl-c 2 sin 2 ≠
than unity. Thus he reduces the general elliptic integral √- —, with abstraction from an algebraic part, to the’
Vl-c 2 sin 2 ≠
three normal forms of “elliptic functions or transcendents ”—
∕'⅛=F, ∕∆d≠ = E, /1 dφ „ . —∏,
√ ∆ y √(l + nsιn 2 ≠)∆
Δ being au abbreviation for the radical Vl - c 2 sin 2 ≠.
With this reduction to fixed normal forms the foundation of the theory of elliptic integrals is laid, and the essentially irreducible integrals found which belong to a square root of a biquadratic function. The same reduction subsequently led to the division of the general Abelian integrals into those of the first, second, and third kinds, in accordance with the properties of these three classes of integrals, either of remaining always finite, or of becoming infinite, algebraically only at infinity, or logarithmically at two different points.
It will be perceived that the epithet ‘ ‘ elliptic” applied to these integrals is purely conventional, arising from the connexion of one of them with the arc of an ellipse; but even at this stage it is apparent that we are concerned with matters of much greater generality than the name indicates. It may also be noticed that, though Legendre calls by the name elliptic functions what are now called elliptic integrals, this is a change introduced by Jacobi, which Legendre long resisted. The change consists in regarding the superior limit of the integral of the first kind as a function of the integral, the latter being now considered as the independent variable. Expressed in symbols the change is that, in Legendre’s equa-
∕∙φ
tion F(»c, φ')=∕ — —-. = u, Jacobi calls ≠=am(w, κ), and sin φ, Jq ∆{κ,φ) cos φ, or ∆φ, &c. (or, in this notation, sin am u, cos am u, ∆am at, &c.), are his elliptic functions.
207. Legendre proceeds, after classifying the integrals, to the comparison of his elliptic functions of the first kind. All geometers, he says, are acquainted with the complete algebraic integral given by Euler of the differential equation
⅛ + ⅜ =0 .
(α + βx + yx 2 + ^x i + ex i )i (a + βy + γy' i + Sy 3 + ey 4 )* ’
the discovery of which, in the introduction, he too ascribes to a combination of good fortune “quoique ces hazards n’ arrivent qu’à ceux qui savent les faire naître.” Our reductions show that this equation can, without loss of generality, be put under the form , √≠ dψ ~ q .
Vi - c 2 sin 2 ≠ Vl -c 2 sin 2 ψ ’
and then its integral is
F(≠)+F(*)-F(μ), μ being an arbitrary constant. But the integral found by Euler’s method is thus written cos φ cos ψ - sin φ sin ψ Vl - c 2 sin 2 μ=cos μ , which he then verifies a posteriori.
The expressions sjn j u≈ ein ≠ ∞s ψ ∆ψ + sin ⅜ cos ≠ ∆≠ t 1 - c 2 sιn 2 ≠ sin 2 ψ ’
c0a cos <∕> cos ψ - sin φ sin ψ ∆φ ∆ψ
1 - c 2 sin 2 ≠ sin 2 ψ ’
_ ∆φ∆φ - c 8 sin φ sin ψ cos φ cos ψ
1 - c 2 sin 2 <∕> sin 2 ψ
are at once derived from this form of the integral; and the corresponding formulae for the amplitude of the difference between two functions follow by replacing ψ by - ψ.
Legendre next proceeds to the formulae for finding algebraically a multiple function of a given one, connecting the angles φ n -ι, φ n , φn+ι, by the relations equivalent to F(≠ n +ι) = F(≠ n ) + F(≠), F(≠n-ι) = F(<∕>,,) - F(≠), which he writes
• , , , 2∆ cos φ. sin φ n alj ≠"÷ 1 + 81nit - 1 - 1 - e ' si n⅜ s i,⅜.
∞ s ⅛⅛⅛-,√ ⅛,.
1 - c a sιn 2 <∕> sιn 2 ≠, 1
These can be applied in succession. Investigating the division of a function into n equal parts the equation is found to rise in general to the degree ri 2 ; but for the complete function the equation is only of the degree ⅛(zt 2 - 1) when n is odd.
208. Proceeding to generalization of Euler’s addition theorem,
Legendre admits that, denoting the radical by R(ae), &c., the equation 0 = + + &c·» can for integer values of m, n, p
R(z) Rtø) R(s)
always be expressed in the form F(μ)=w,F(≠) + nF(ψ) + &c., and so will always have a complete algebraic integral, for nothing prevents the supposition that z and the following variables are given algebraic functions of x and y. Perhaps, he says, this is the only way of generalizing Euler’s result concerning the equation
V? ’ β⅛ ∕
RÜÖ + I r (yj = θ ’ F° r, Lagrange tried to find cases of integra-
,. 1 .. „ dx dy bιlιty of + , without supposing the two polynomials X
and Y entirely similar,· it does not seem that he arrived at any any other result; the equation he gives (Mém. de Turin, iv. 119) is immediately reducible to Euler’s. Thus, as has been remarked, Legendre was at this time very far from anticipating the very general transformations, since discovered, or the celebrated theorem of Abel which so marvellously extended this subject.
209. Having illustrated the functions F by the lemniscate and other curves, algebraic and transcendent, whose arcs are expressed by functions of the first kind, as well as by the expression for the time in the motion of a simple pendulum, Legendre enters, in chapter ix., on the comparison of elliptic functions of the second kind. Corresponding to the relation F(≠) + F(ψ) - F(μ) = 0, these functions are related by
E(<∕>) + E(ψ) - E(j u) = c 2 sin μ sin φ sin ψ.
This includes Fagnani’s as a particular case, and of course there is a similar relation for comparison of the arcs of hyperbolae. In chap. xii. the well-known relation of Legendre is established between complete integrals of the first two kinds with complementary moduli; b and c being moduli are said to be complementary when δ 2 +c 2 = l. Denoting by F, E the values of F c^, E^--J-, and by F', E' those of , E^-^-,δJ , this relation, which has been already demonstrated, § 183, is
FE'+F'E-FF'∙=-^-. Δ
These complete functions satisfy differential equations of the second order, viz., F satisfies dc 2 c de ’ and the complete E
(l-c 2 )≤f + ≥l≤ ⅛=0, de 2 c de with corresponding equations when b is taken as the independent variable. The complete integrals of these differential equations are assigned in terms of both sets of complete functions, and the differential equations are utilized to shoλv the law of the development of these functions in series of powers of the complement of the modulus, since when the modulus is near unity the ordinary series in powers of the modulus do not sufficiently converge.
210. In treating integrals of the third kind, the presence of a third determining magnitude, the parameter n, besides the amplitude φ and modulus c, which are common to the first two kinds, is an additional complication. Legendre first establishes the relation
Π(n) + ∏(⅛)-F + * tan -.√j⅛i⅛, ∖ n ∕ Vα ∆
where α = (l+n)^l+; by means of which any function ∏ having a parameter greater than c is reduced to depend on one having a parameter less than c, but with the same amplitude and modulus. The quantity a, however, may have different values, and thus the following cases are to be distinguished.
When a is positive (either n positive, or if negative its value is between -1 and - c 2 ) the function introduced is circular, as written above.
When α is negative, n is negative, and either greater than -1 or less than -c 2 , and the function is logarithmic. In this case writing α = - β, the comparison written above is
∏(π) + ∏(-) = F + — l r log (A + ^ tan *\ , ∖ n J 2√B 5 ∖Δ - √β tan φ∕
When α = 0 the integrals are expressed by the first and second kinds.
Omitting the case of n= - cosec 2 0, which can be reduced to that of n = - c 2 sin 2 θ, this case and those of n = cot 2 0 and n = - 1 + b 2 sin 2 0 remain, the first being the logarithmic parameter. The other two cases are easily shown to be connected by the relatipn
1±⅛) - ∏( - m ) = — + -jL= tan -1 V^∙sin≠cos0 n m rnn ∖ f m n a ’ provided (l + n) (1—∞) = δ 2 , and so constitute really but one case. Functions with imaginary parameters always reduce to others with real parameters of the above two distinct kinds.
211. Comparison of integrals of the third kind by means of the addition theorem leads to the formula
∏(ψ) + ∏(f )-∏W=ψtan-* j n√≈ si∏ M sin ≠ sin ψ) .
Vα ( 1 + n-n cos μ cos φ cos ψ ) ’ and thus the difference, which is zero in the first kind, and is algebraic in the second, is here expressed by the arc of a circle; which becomes a logarithm if a be negative. Thus finally Legendre remarks that if J √(α + βx + ya? + δx 3 + ∈Z 4 ) ’ where P is a rational function of x, there can always be found an algebraic equation between x, y, z, &c., such that the quantity iZ(x) + kZ(y) + lZ(z) + &c., where i, k, l, &c., are integers, may be determinable by arcs of circles and by logarithms.
212. Legendre next proceeds to the discovery of Landen, having so far been employed mainly with those of Euler. He expresses astonishment that among the many analytic transformations employed by Maclaurin and D’Alembert they had not fallen in with the transformation which brings to light the numerous properties of the chain of moduli, and that this discovery was reserved for Landen, who, however, made but a poor use of it, not even seeing that it furnished a very simple method for approximate calculation of the arcs of conics. It is less astonishing that Euler missed this discovery considering that the beautiful integration which is due to him led himto compare together the different values of the same transcendent, just as arcs of the same curve are compared. But nowhere in his Mémoires do we find him varying the constants or parameters of the functions, and thus passing from one curve to another, as is done in comparisons which depend on the scale of moduli. From the fact that Euler has written nothing about the memoir of Landen, Legendre concludes he had never been acquainted with it.
213. The formula given in § 194, by introducing the eccentric angles from the axes minor in the two ellipses, easily gives rise to the equation sin ≠ 0 ≈(1 +⅞) s ^ n cos ≠, w } le re δ=- , and so
Δ m c = Vÿ of that article. ≠ 0 belongs to another ellipse, and for it the value of the corresponding modulus is evidently . Legendre sees through the simple proposition of Landen, expressing the arc of a hyperbola by two arcs of ellipses, to the infinite series of moduli which can be produced by repeated application of this substitution. Commencing with functions of the first kind, he shows that F(c,≠) and F(c',≠'), when c, ^=y~~, and φ' is determined by sin (2<∕>' - φ)==c sin φ, are related by the equation
F(< ≠') = ⅛⅛, ≠).
Thus, as 2φ ’ - φ is always contained between + 0 and-Ø, ø being the least arc having c for its sine, there is no ambiguity in determining the values of φ' and φ. The relation for complete functions is F 1 (c) = (l +c)F 1 (c). Now conceive an infinite series of increasing moduli
, 2√c j, 2√c ’ 2√√'.
1+c 1 + √ l+c" it will soon attain the limit unity. Forming the complements b', b", &c., of these moduli, the series decreases continually, and each term is, according to the law,
, 5"=≥≡X , , &c·
1 + c’ 1+c’ 1 + c”
The series of amplitudes is deduced in succession by the formulae sin (2φ ’ -φ)=c sin φ, sin (2φ"-φ')=c ’ sin φ ’ , sin (2φ , " - φ") = d' sin φ'', &c.; and so a series of integrals of the first kind is got, related as follows F(< ≠')=⅛F(c,≠), Δ
F(c", ≠") = l+i ’ F(c', φ') =l±- c ⅛-'f(c, ≠), &c., Δ Δ Δ any two of the functions being always in a ratio independent of the values of the corresponding amplitudes. Similarly for the complete functions
F 1 (c ’) = (l+ c)F 1 (c), F 1 (c")≈(l+ c ’)F 1 (c') = (l+c)(l+ c')F 1 (c), &e.
But the series c, c', c", increasing in one direction, can be prolonged indefinitely in the opposite or decreasing sense to the limit zero. Here the law of terms is
2√c δ ^ 0 2√c u0 00 2√c≡ , c ~l+c° , C= l + c 00 ’ C ι + c ooo∙ &c>; and similarly n 1-5 00 l-δ 0 .
c ~ι+b ,c ι + δ° ,fcc ∙ ’ with the series of relations of amplitudes derived from sin (2≠ - ≠ 0 ) = c 0 sin φ 0 , &c., which may be written tan (φ 0 -ψ') = b tan φ, &c., and of integrals F(c, φ) = ^^ c F(c 0 , φ 0 ), &c.; Δ 2 or, reversing, since 1 + c ° = j^_^ »
Γ(c 0 , ≠θ)=≈(l + t)F(c, ≠),
F(c° 0 , φ 00 ) = (1 + δ 0 )F(c 0 , ≠ 0 ) = (1 + δ)(l + δ 0 )F(c, ≠), &c., and, for the complete integrals,
W)-l±⅛(c),
F 1 (c 00 ) _ l+l 0 f i( c 0) = 1 + δ L+ÈÜP(c), &c., Δ Δ Δ for the decreasing scale of moduli.
214. Now if this be applied to the second kind of integrals we find
J 2 F(c, ≠) = 2E(c, φ') - 2(1 + c)E(c ’, φ') + 2c sin φ, showing that an integral of the first kind can be expressed by the aid of two arcs of ellipses, E(c, φ) and E(c', φ'). Whence by the formula of § 192 it follows that an arc of a hyperbola can always l∣e expressed by two elliptic arcs, the theorem Landen enriched geometry by. Also, by eliminating the integral of the first kind by means of two consecutive equations in the series, the relation between the arcs of three consecutive ellipses in the series can be found; so that by the indefinite rectification of two ellipses in the series
. . . E(c", φ"), E(c', ≠'), E(c, φ), E(c 0 , φ 0 ), E(c 00 , ≠ 00 ), . . . of which the extremes are the ellipse having an eccentricity 1, which is a portion of the axis major, and that having an eccentricity 0, which is a circle, the rectification of all the rest is obtained.
The transformations of Lagrange, or of Gauss (§ 203), may be seen to be essentially the same as this of Landen (or Legendre), for by taking sin + in F(c, φ) we get x - r Λ» Sill ψ dφ (l+fc)dψ , 2 ∖ fk . 0 r. w here c = -— i - , or k = c 0 .
V1 - c a sin 8 ≠ y fl-k 2 sin 2 ψ 1 + »
Hence F(c, ≠) = (1+c 0 )F(c 0 ,ψ).
Now with F(c 0 ,ψ) = ⅜F(c 0 ,≠ 0 ), F(c, ≠)≈ 1 ^^ c ° F(c 0 ,≠ 0 ), andby elimi-
• · · (1 i cθ) sin. ιl∕ nating the quantity ψ between the equations sin φ = θsin 2 ψ ’ tan ⅜≠ 0 = tan ψ√j _ c 02 s j n 2ψ j we obtain the relation given above sin(2≠ - φ 0 ) = c 0 sin φ 0 .
215. This principle of transformation is next applied to the approximate calculation of the three kinds of integrals. Required, for instance, an approximate value of F(c, φ): the decreasing moduli c 0 , c 00 , c 000 , . . . must be calculated, and the increasing amplitudes ψ 0 , φ 00 , φ 000 , . . . thus
F⅛ ≠)≈ 1 -±L°F( c o, ≠o) - 1±L° 1 -±≤⅛θ<>, ≠θ°) = &c.; Δ ΛΛ Δ but, when the modulus has become very small, ∆ = l, and/*— = ø; if then Φ be the limit of the angles ⅜≠ 0 , ⅛φ 00 , ⅜≠ 000 , &c., we have
F(c, ø) = Φ(l + c 0 )(l + c c0 ). . .
When <∕>=⅛ιr the limit Φ will be equally ⅜τr; so that the complete function is
F 1 (c)=⅜π(l + c 0 )(l + c 00 )(l + c 000 ). .
The continued product which multiplies ⅜π, or Φ, may also be written in a form suited to logarithmic calculation, as √ jojoo⅛ooo b
The moduli are best got by taking auxiliary angles: let sin μ=c, .∙. c 0 =tan 2 ⅜μ, similarly if c 0 =sin μ 0 , c 0 ° = tan 2 ⅛μ 0 , &c.; and when a very small c has been arrived at, we can get the next by c 0 = 1c 2 + i.∣c< + ⅜.⅜.ic β + . . .
Also the angles φ 0 , φ 00 , are best found by tan (ø°- 0) = δ tan φ,— taking for φ l1 - φ not always the least angle given by the tables but that which is nearest φ.
216. Combining the equation Tl = J r v .∙ . with that de-
J (l+nsιn 2 0)∆0
rived from it by differentiation with regard to n, and using α to denote (1+ n) + as in § 210, it is easily found that , . p du 913 p (lu f^∏y pdn
2√α. ∏=Δsm φ cos φ∕ . . .- -cF∕ - (f -E)∕—— .
(l+nsιn 2 0)√α y π 2 √a J n√ a Applying this to the case n =cot 2 0, and writing for brevity cot 2 ⅞5= n', the following relation is found :—
x> π (w> c,Φ) + -t- δ ^ >∏(√, δ, θ) sin 0 cos 0 v ' sin φ cos φ ’ ’ ’
= ⅜ιr+tan 0∆(δ, 0)F(c, φ) + tan φ∆(c, φ)F(δ, 0)
+ F(c, φ)F(δ, 0)-F(c, φ)E(δ, 0)-E(c, φ)F(δ, 0).
Making φ=⅛τr, this gives for the complete function of the third kind, with positive parameter, the following expression :—
+ F 1 (c)F(δ, 0)-F 1 (c)E(δ, 0)-E 1 (c)F(δ, 0).
A similar relation is established for the other cases of the parameter, and in each the complete integral is likewise expressed by integrals of lower kinds.
There now follow the general reduction of integrals with imaginary parameter, and the reduction to elliptic integrals of integrals not included in the general type, as for instance √ c dz p dφ p dφ
(1+jpz 2 )(l + g ’ z 2 )⅛ y (1 - c 2 sin 2 0)i J (l-c 2 sin 2 φ)i ’ °*
217. In his preface, however, Legendre had directed attention to the discovery of a new scale of moduli, different from that hitherto known, as the most novel of the results distinguishing this work from his Exercices. This transformation starts from the assumption sin φ(w + ⅞ sin 2 φ) e l+⅛sin 2 φ ’
and by the conditions that a> and φ reach ⅜π together, and, moreover, that cos ω does not contain any other irrational factor in sin φ but cos φ, we get cosω ° c08 rl 1 ⅛~s⅛⅛ ii ^ , λ=⅛(∞-1) 2 ∙
JL ^τ^ κ bill φ
Now da>≈ d Φ( m ~ s * n ≠) hence tan ∣(ω + φ)=tan ø. l+⅛sιn 2 φ ‘ 2
But in order that ω should increase"gradually from 0 to ∣τr, as φ does, h must be less than 1, and m less than 3.
Again, if c and a are two moduli, so related that
1 sin 2 φ ∖2
1 , T · a ∕ , l+⅛ sιn 2 φ∕ ’
it will be found possible in general to satisfy the above equation, and thus we get
^ Qra-l)(w+3) 8 and r , a (τn-l) 3 (rn + 3).
16rn 8 ’ = 16wι ’
whence m must be between 3 and 1 in order that a and c may both be real proper fractions.
k
1-—sin 2 φ
Hence ∆(α, ω) = -z—~·. a ∆(c,φ); v, , l + ⅛sιn 2 φ κ,ψ ' ’ and this, combined with the above differential relation, gives dω mdφ ∆(,a, ω) ∆(c, φ) ’ or, integrating, F(α, ω) = rnF(c, φ),— a relation between two functions of the first kind, whose moduli depend in general on the quantity m, which may be taken at will between the limits 1 and 3.
The modulus, a is always greater than c, for we have
⅛ β ^wιτF an< ^ m + '~ ’ ~ (m2 -wι ) = (l + wι )(3 -∞), which is always positive. We have seen that α and c are determined by means of the regulator m when it is known; it can be found from either of them by solving a biquadratic.
Again, the complements of the moduli are found by 2 (rn + l)(3-rn) 8 (τn + l) 3 (8 -m) p 16rn 8 ’ = 16rn whence follows the simple relation
√0c+√j8δ=l.
Application of this transformation to integrals of the second and third kinds gives rise to the remark that the trisection of an indefinite function of the first kind may be reduced to depend on the solution of two cubic equations.
218. Now, starting with a given modulus c, an infinite series of moduli increasing towards the limit 1, and an infinite series decreasing towards the limit 0, may be formed, and we may denote the latter by a notation analogous to the former. Let them be in the increasing order c, c⅛, c u , c ιπ , &c., and in the decreasing order c, c o> c oo> &°· > ar *d similarly for the complements, the regulators, and the amplitudes. Thus, by the first scale, any integral of the first kind, having a given modulus and amplitude, can be transformed into another with any modulus in the series . . . c 00 , c 0 , c, cl, c". . . and from this by the second to any in the other series formed from the same c by a different law, depending on extractions of square and cube roots.
Legendre arranges the moduli in a sort of infinite chess-board, having c in the centre, and the moduli derived according to each scale in rectangular directions, and notices how remarkable is this infinite multitude of transformations which the same function F(c, φ) may be submitted to, without changing its nature while preserving the same ratio between the new function and the old for all values of the amplitude; in vain, he adds, might a second example be sought of a function which should be reproduced under so many different forms, and to which, more justly than to the logarithmic spiral, might be applied James Bernoulli’s device, “Eadem mutata resurgit.”
219. The first volume of the Traité also contains the reduction to elliptics of a great number of integrals, the development of elliptic integrals in series proceeding by sines and cosines of multiples of the amplitude, and calculations of some definite integrals, single and double, which can be expressed by elliptic integrals. The applications are, in geometry to the surface of an oblique cone, to that of an ellipsoid, and to a geodesic on a spheroid; and in mechanics, to the rotation of a solid, to the motion of a body under the attraction of two fixed centres, to the attraction of homogeneous ellipsoids, and to the orbit described under a given central force.
The second volume contains details of the calculation of the integrals, and such tables of them as have to be constructed in order that the use of these functions may be introduced into analysis just as circular and logarithmic functions are employed. Here, Legendre excludes the thought of reducing to tables functions of the third kind, since they contain besides the principal variable two arbitrary quantities; and so the tables should be of triple entry, a thing altogether unmanageable. Besides these, this volume contains a treatise on Eulerian integrals, and an appendix on spherical functions and on quadratures.
The third volume of the Traité contains three supplements to the theory of elliptic functions, dated 1828, 1829, 1832, embodying Legendre’s acceptance of the discoveries made by Jacobi and Abel since the publication of the Traité.
220. It was owing to the strangeness of his subject that Legendre for more than twenty years found no fellow-worker in it. “After having employed myself for a great number of years,” he says in the preface to the first supplement of the Traité, “with the theory of elliptic functions, of which the immortal Euler had laid the foundations, I thought I should collect the results of this long work in a treatise, and this I published in the month of January 1827. Up to that geometers had taken almost no part in this kind of researches; but hardly had my work seen the light, its name could hardly have become known to scientific foreigners, when I learned with equal surprise and satisfaction that two. young geometers, MM. Jacobi of Königsberg and Abel of Christiania, had. succeeded by their own studies in perfecting considerably the theory of elliptic functions in its highest parts.”
Abel and Jacobi have found not only points of connexion for their works with Legendre’s investigations, but have been able to adopt a set of methods and points of view from his Traité, on the basis of which they have constructed the mighty edifice of the theory of elliptic transcendents. This Jacobi himself subsequently fully recognized. On May 27,1832, he writes to Legendre “In a note at the end of the eighth volume of Μ. Grelle, I have sought to extol the imperishable merits of the geometer who, besides the numerous and important discoveries with which he has enriched science, has effectually laid the foundations by the glorious labours of his life of two great and extended disciplines which shall henceforth form the α and the ω of every mathematical study. I have at the same time made use of this opportunity to speak of Abel and of his great theorem, which you again have the merit of having first penetrated, and of having shown to posterity that its development is the great task remaining for it to fulfil.” . -
221. Before entering into the developments due to these two celebrated mathematicians we should make some mention of Gauss’s labours in the same field. These were mostly found in incomplete sketches at the beginnings of different treatises, or as individual formulae scattered among his other works. The editor of his collected works has brought them together in the latter part of the third volume, and states that there is evidence that Gauss was acquainted with the relations between the arithmetic geometric mean and the series proceeding by square powers in the year 1794. This arithmetic geometric mean is defined in the paper “Determinatio attractionis” published in 1818, where he speaks of it as Λ peculiar and most expeditious algorithm which he had for many years employed, and intended yet to treat of more fully. __
Let m, n be two positive quantities, andput nι ’ =⅛ {m +-π), n'— ∖ Jmn, so that m' and n' may be their arithmetic and geometric mean, taking the latter always positive. Now take
τn"=⅛(m' + ri), n"=√m'n' > m"' ^⅛(m"+n"), n"' = V m"n" > and so on. It may be seen that the series m, m', m", m"', and n, n', n", n'", &c., rapidly converge to a common limit, which we shall denote by/*, and simply call the arithmetic geometric mean between wιand n. Now wo shall demonstrate that A is the value of the μ ∙ integral √ 2 dr
2π∖∕ wi s cos 2 τ+w a sin 2 τ
taken τ = 0 to τ=∙330°.
For suppose that the valuable τ is expressed by another τ', so that 2m sin r'
*jm - * * " —-≡*-i. μ. . i. » !∣
(zzι + n) cos 2 τ' + 2m sin 2 √
it will easily be seen that as τ' increases from 0 to 90°, 180°, 270°, 360°, τ also, though not uniformly, increases through the same range. But, effecting the substitution, ^
dr dr .
√(m 2 cos 2 τ+ii 2 siιι 2 τ) √(w a cos 2 √ + 7i' 2 sin 2 -r') ’
accordingly the values of the integrals / “ dr f dr 2rt ∖ J(rn^ cos 2 τ+n 2 sin 2 τ) t J 2r√(w 2 cos 2 √ + √ 2 sin 2 τ'). ’ each taken from 0 to 360 β are equal; and, since this may be carried on as far as we please, plainly they are also equal to the value of the integral Γ dΘ J2πJ(μ 2 cos 2 Θ + ∕* 2 sin 2 ©) ’
from Θ = 0 to Θ = 360°, which is plainly — .
P 2 22. N. H. Abel (1802-29) started in the summer of 1825 to pursue his studies abroad, chiefly at Paris. On his way he made the acquaintance in Berlin of A. L. Crelle, who had long conceived the project of founding a mathematical journal, but was decided to put this into execution by the importance of the numerous memoirs already prepared by Abel (as also by Steiner), who consented to their publication in it. The first number of the journal appeared during Abel’s stay in Berlin, and each copy in the first four volumes contained papers by him. These and other published papers are reprinted in the first volume of his collected works (Christiania, 1839). The second volume contains mostly papers found after his death, nearly all in this volume having been written before. Abel began his travels. These., therefore, first claim our attention after Legendre’s independent investigations.
223. Elliptic integrals have hitherto occupied us. We have mentioned (§ 207) the problem of inversion which leads to elliptic functions, viz., that if w=F(κ, <⅛), then ≠ = am u, and if Γ x dx , u=∕ -τ≈= , we have x= sin am u. Jo √(l-α 2 )(l--r⅛ 2 )
Paper viii. (vol. in), is headed “Remarkable properties of the function deteπnined by the equation fy.dy-dx{(a-y)(a 1 -y)(a. 2 -y) . . . (a m ~ y)} 4 -0, fy being any function of y which does not become zero or infinite when y==a, a 1 , α 2 , . . . a m .” In it this problem of inversion of the more general (hyperelliptic) integral in which the square root contains a function of the degree m +1 is attempted; and, though it has since been shown by Jacobi (Grelle, xiii.) that the inversion of hyperelliptic integrals is a different problem from what is here proposed, Abel at any rate in this short paper had established the existence of two periods for elliptic functions.
224. We saw in § 216 that Legendre, in comparing two elliptic integrals of the third kind, found a certain relation connecting with integrals of the first and second kinds two such integrals of the third kind, in which the argument and parameter are interchanged. This relation recurs to us in an extended form in the paper of Abel (ix., vol. ii.) “On a remarkable property of a very extended class of transcendent functions.” Defining y or ψx by the differential equation y . fx ⅛ . ≠ic∙=0, where φx≈a + a 1 x + a. 2 x 2 + . . ., and∕ε=3 + 3 1 aJ + β 2 iε 2 + . . . , he gets
1 fψx. dx l f da ψa J * x ∙ φ J(ax)ψa . ψα
= S((n+ 2)αm + n +2 - 'fi^⅛ xdy o the integrals in x being taken from a value of x which makes φx . φx vanish, and those in a from a value of a which destroys A. When we put ψ¾° r--1=- in this, it becomes the theorem for the ψα ∖ ∕φx interchange of argument and parameter for hyperelliptic integrals.
225. “The first works of Abel which attracted attention,” writes his editor, “were his memoirs on the impossibility of the general resolution of algebraic equations higher than the fourth degree, and his researches on elliptic functions. Simultaneously with Abel, and without being acquainted with his works, Μ. Jacobi of Königsberg began to treat the theory of elliptic functions. Thus a rivalry exists between these two men of exalted genius in their treatises on these functions. Abel told me that during his stay at Paris in 1826 he had already completed the essential part of the principles he subsequently enunciated regarding these functions, and that he would have much wished to postpone the publication of his discoveries until he could compose a complete theory of them, had not in the meanwhile Μ. Jacobi entered the lists.”
226. On October 24,1826, Abel wrote from Paris:—“I have just finished a large treatise on a certain class of transcendent functions for presentation to the Institute, and that will take place next Monday. I dare without ostentation say it is a treatise which will give satisfaction. I am curious to hear the opinion of the Institute about it.” He had not deceived himself in the significance and reach of this fundamental theorem; yet in the Academy judgment upon the work was deferred, so that Abel two years later (Jan. 6, 1829), felt himself called upon to send to Grelle the following, which appeared in the fourth volume of the Journal ( Works, i.p. 324).
227. ‘ ‘ Demonstration of a general property of a certain class of transcendent functions. “Theorem. — Let y be a function of x which satisfies any irreducible equation of the form ^>=Po+Pιy+Pif+ · · · +pn-ιlt n ~ l +f i · · · (i), where p w p 1 , . . . p n -ι are integer functions of the variable x. In like manner let q 0 , g 1 , . . . g n -χ be integer functions of x, and
θ≡9'o+9!S' + S'2r 1 + · · · +i∙-ιy n ^ 1 · · · · ( 2 ) a similar equation, ànd let us suppose the coefficients of the different powers of x in these functions variable. Let these be denoted by a, a!, a" . . . . By reason of the two equations (1) and (2), x will be a function of a, a!, a! , , &c.; and we shall determine its values by eliminating y. Let us denote by
P = 0 (3)
the result of elimination, so that p will contain only the variables x, a, a , , &c. Let ∕t be the degree of this equation in x, and let its μ roots be ael, X. 2 , . . . Xμ (4), which will be so many functions of a, a!, a", &c.
“Now, if f(x, y) denote any rational function of x and y, and we make φx≈H(x, y}dx (5), then the transcendent fπnction ψ(x) will possess the general property expressed by the following equation—
ψaq + ψx 2 + . . . ψx μ *=u+fc 1 logt⅛+⅛ 2 logv 2 + . . . + k n ∖ ogv n .. . (6), u, v ι> v 2 , . . . v n , being rational functions of a, a', a", . . . and Λ⅛, k i , . . . k n , constants.
it Demonstration.— To prove this theorem it is enough to express the first member of equation (6) as a function of a, a!, a", &c.; for thereby it will reduce to a rational differential, as we shall see.
“First, the two equations (1) and (2) will give y as a rational function of x, a, a', a'. . . Similarly the equation (3) will give for dx an expression of i he form dx≈ uda + a'da'+a"da" + &c., where α, a', a" . . . are rational functions of x, a, a!, a!', &c. Thence it follows that the differential f{x, y}dx can be put under the form f[x,y)dx≈φxda + φ i xda'+φ 2 xda"+kc., where φx, φ 1 x, ... are rational functions of x, a, a', a", &c. Integrating, we get φx=f'(φxda + φ 1 xda'+ . . .); and from this we conclude, since this equation holds when we put for as its μ values,
ψx 1 + ψx 2 + . . . +φx μ ≈f(φx 1 + φx. i + . . . + φx μ )da ) . 7 .
+∕(≠jX 1 + ≠ 1 x 2 + . . . + φ 1 x μ jda'+L·c. ∫
“In this equation the coefficients of the differentials da, da ’ , &c., are rational functions of a, a’, a" . . . and of x 1 , x 2 , . . . x μ ; but they are besides symmetrical in x 1 , x 2 , . . . x μ ; therefore, by a well-known theorem, they can be expressed rationally in terms of a, a', a" . . . and the coefficients of the equation p = 0; but these latter are themselves rational functions of the variables a, a', a" . . . so that finally the coefficients of da, da’, da", &c., of equation (7) will be so too. Therefore, integrating, we have an equation of the form (6).
“I propose on another occasion to develop numerous applications of this theorem, which will throw a great light on the nature of the transcendental functions it deals with.”
228. Abel died of consumption, April 6, 1829, having been confined to bed nearly three months; and. of the applications promised nothing appeared or has since been found in his remains. Moreover, except the note that he had presented the memoir to the Academy, which appears in the paper t ‘ On some general properties of a certain sort or transcendental functions” (Works, 1. p. 288), Abel hardly seems to have expressly referred to it, though he mentioned the theorem (November 25, 1828, Works, ii. p. 258) to Legendre, adding that on this general property he had in fact founded the whole theory of elliptic functions.
229. But notwithstanding, his contemporaries were not slow to estimate the value of his analysis. The statement in Crelle , s Journal revealed to Jacobi the entire significance of this “fundamental theorem of analysis,” and his admiration breaks forth to Legendre on the 14th March 1829 :—“What a discovery of Abel’s is that generalization of Euler’s integral? But how does it happen that this discovery, perhaps the most important our century has made in mathematics, though communicated two years since to the Academy, has been able to escape the attention both of you and your fellow members?” To this question Legendre answers (April 8, 1829) :—“I shall not close this letter without answering yours relative to Μ. Abel’s beautiful paper, which was printed in the last number of CrcUe’s Joυ∕rnal, and which had been presented to the Academy by its author in the last months of 1826. Μ. Poisson was then president of the Academy. The committee named to examine the memoir were Μ. Cauchy and myself. We perceived that the memoir was almost illegible; it was written with very faint ink, the characters badly formed; it was agreed on that we should ask the author for a better copy, and one easier to be read. So things remained. Μ. Cauchy kept the manuscript up to this without doing anything further about it. The author, Μ. Abel, appears to have gone away without caring what became of his memoir. He furnished no copy, and no report was made. However, I have asked Μ. Cauchy to give me the manuscript, which never was in my hands, and I shall see what there is to be done, to make up, if possible, for the little attention he bestowed on a production which no doubt deserved a better fate.”
230. The third and last supplement of Legendre to his great work is dated March 4,1832, and concludes as follows :—“Here we shall terminate the additions we proposed to make to our work, taking advantage of the recent discoveries of MM. Abel and Jacobi in the theory of elliptic functions. It will be remarked that the most important of these additions consists in the new branch of analysis we have deduced from the theorem of Μ. Abel, and which had hitherto remained quite unknown to geometers. This branch of analysis, to which we have given the name ‘ theory of ultra-elliptic functions,’ is of infinitely greater extent than that of elliptic functions, with which it has very intimate relations; it is composed of an indefinite number of classes, each of which divides into three kinds, as do elliptic functions, having besides a great number of properties. We have only been able to glance at this matter, but no doubt it will be gradually enriched by the labours of geometers, and at length will form one of the finest parts of the analysis of transcendents.” At nearly the same time Legendre wrote to Crelle— “The work, so far as I am concerned, has given me the profound satisfaction of rendering conspicuous homage to the genius of Μ. Abel, in making felt all the merit of the beautiful theorem which was his discovery, and which maybe characterized as Monumentum aere perenniusB In his remarks on this third supplement of Legendre {Grelle, viii.) occur the notable words of Jacobi :—“We hold it (the Abelian theorem) to be the greatest mathematical discovery of our time, although it remains for a future, perhaps long distant, work to manifest its whole significance.”
231. The name which Jacobi thus applied, the Abelian theorem, has since adhered, and the functions to which it refers have been called Abelian functions, the term hyper-or ultra-elliptic having been restricted to that particular class in which the square root of a polynomial is the only irrational quantity introduced; while Abelian functions may depend on any irrationality. The neglected paper of Abel appeared in the Memoirs oL the Institute in 1841.
232. But, though the Abelian theorem was thus published during its author’s stay in Paris, his labours in other departments of the theory of transcendents suffered no intermission. In December 1826 he writes—“I have written a large memoir on elliptic functions which contains much that is curious, and which I flatter myself will not fail to fix the attention of the literary world. Amongst other things it treats of the division of the arc of the lemniscate. I have found that with rule and compass the lemniscate can be divided into 2' , + l equal parts, when this number 2 n + l is prime. The division depends on an equation of the degree (2^ + l) 2 -l, but I have found its complete solution by means of square roots. This has revealed to me at the same time the mystery in which Mr Gauss’s theory of the division of tho circumference of the circle has been wrapped up. I see clearly how he arrived at it,” referring to tho last section of the Disquisitiones Arithmeticae of Gauss, published in 1801.
233. G. G. J. Jacobi (born Dec. 10, 1804, died Feb. 18, 1851) appeared first as a discoverer in connexion with our subject in the “Extracts of two letters to the editor,” published in September 1827 in the Astronomische Nachrichten of Schumacher, and reprinted at the beginning, of the collected Works, 1881. We have seen that Legendre discerned the vast importance of the relations which he called Landen’s transformation, and discovered with increased wonder the further transformation of the third order, which became public in January of this same year 1827 in the Traité. But in his first letter Jacobi states: “The integrals of the form ∕*. f fy* for different moduli c, belong to different J √1 - c 2 sin 2 ≠
transcendents. Only one system of moduli is known for which they reduce into, one another, and Μ. Legendre in his Exercices says even that there is only this one. But in fact there are as many of these systems as there are prime numbers, that is to say, there is an infinite number of these systems, all independent, each answering to a prime number; the system heretofore known answers to the prime number 2.”
This is already, without proof, α statement of the general theorem of transformation of elliptic integrals of the first kind. If U be a certain odd function of sin ψ of any degree n, and V a certain even function of sin ψ of degree n -1, then, putting sin ≠=-γ, the coefficients of these functions may be determined so as to satisfy 7” dφ f dψ .2 ==≈g¾ ’ ∕ .7√ ι ∙,Λ .^V. > and each of these substitu- √1 - c a sιn i ⅛> 7 √1 -⅛ s sιn 2 ψ ’
tions gives a new system of moduli.
Further, Jacobi notices that sin ψ can be in an almost analogous manner expressed by sin θ, so as by composition of the two integral equations to satisfy the relation f clφ ∕* dθ
√Vl-c a sin⅛∙ yVl-c 2 sin 2 θ
Tlιus the substitution which serves to give n times the transcendent can be divided into two of a simpler nature, and this substitution gives sin φ expressed by a fraction whose numerator contains the odd powers of sin θ up to n 2 , and its denominator the even powers of it up to n i -1.
Without giving the general proof, the transformations of the third and of the fifth degrees are here actually effected, and connected with multiplication and division for the numbers 3 and 5; and thus for the first time the algebraic solution of the equation of the ninth degree which trisects the transcendent is given.
234. Legendre could not at first believe in the existence of an algebraic transformation belonging to any arbitrary degree, and thought Jacobi trusted to mere induction. But he soon admitted the profundity and rigour of Jacobi’s analysis on receiving from him a letter, dated Aug. 5, 1827, in which it is stated that, if^ be any odd number, we can by a rational substitution, z(A÷A⅛ 2 + . . . +A'⅛P 2 ~ 1 )
X B + B'z 2 + . . . +B"zP t ~ 1 ’ arrive at the equation dx σι dz
√(1-x 2 )(l-κ⅛ 2 j “j7 √(l-z 2 )(l- z Wj ’
Moreover, this substitution can be replaced by two in succession, r _ y(a+a'y 2 + · · · +a"y p ~ 1 ) y = z(a+alz a + , . . + a ,, z p ~ 1 ) X b + b'y 2 + . . . b"y p ~ 1 ’ β + β'z 2 + . . . + β"z p ~ 1 ’ the first substitution transforming the elliptic function into another of different modulus, so that dx ^M,dy √(T→j 2 )(1 - Λ 2 j = √(1 - ι∕ 2 )(l^→V) ’
and the second returning to the original dy p dz √(l-2∕ 2 )(l-λ 2 y 2 )≡M √(1 - z 2 )(l - κ Φj *
Now, givings different values, we see that each given modulus is one in an infinite scale of moduli into which it may be transformed by an algebraic and even rational substitution. This letter, moreover, contained the two theorems giving the general formula in a trigonometric form.
235. Subsequently Jacobi, on January 12, 1828, brought under the notice of Legendre the works of Abel on elliptic functions, which Grelle had published,—but in his own notation. Abel, he says, begins with the analytic expression of all the roots of the equations of higher degrees on which the division of elliptic functions depends. Taking sin φ = i tan ψ, where ι = √-1, and y ⅜ _ f -∕√l-κ 2 sin 2 φ
we have, if κ' be the complement of κ, dφ _ idφ
√1 - κ 2 sin 2 ≠ √1 - κ' 2 sin 2 ψ
whence sin am (zξ, κ)=⅛ tan am (ξ, √), which is a “fundamental theorem of Μ. Abel.”
Further, we have in general sin am (ζ + 4∞K + 4m'⅛K') = sin am ξ, where m and m' are positive or negative integers, and K is the complete function
∕^v dφ JO Vl-κ 2 sin 2 ≠*
and K' the complete function corresponding to κ'.
We see then that the roots of this higher equation for dividing the elliptic function ξ into n parts will be of the form sin am ⅜⅛ κ+wκ 'S
n a formula which involves as many as n 2 roots, if we make m, m' successively take the values 0, 1, 2, . . . n -1.
Abel next reduces the division of any elliptic function ξ to the division of the complete function K. In fact, if α, β be any roots of x n =l, the expression (⅛a m β m ' sin am^^ l ^ , giv ing m, m' all the values 0,1, 2 . . . n -1, will not change if we put instead of sin am any other root, sin am E÷∕mK+⅜∕*⅞K w ∣ ∩ η n thus be symmetrical in these roots, and may therefore be expressed by sin am ξ, and by constant but irrational quantities of the form
4τnK + 4wz7K' sin am .
n
Now giving α, β all possible values produces n 2 combinations, and thereby the values of all the roots. The division of the complete function, which depends in general on an equation of the ⅜(n 2 -1) degree, is reduced to one of the n +1 degree, n being a prime number. For if ω = 4∕*K + 4μ K i , an j g] 3ea primitive root of the congruence x n ~ 1 ≡1 (mod n), also φ (ω) any trigonometric function of the amplitude of co, and. a a root of ic n ~ 1 = l, we attain this by considering the expression
[≠(ω) + aøtøco) + α 2 0(<∕ 2 ω) + . . . + a n ^ 2 ≠(^ n - 2 ω)] n ^ 1 which is symmetrical in
≠(w)> ≠(9 ’ ")> ≠(9 r2 w) · · · ≠(<7 n ^ 2 ω)∙
But symmetric functions of these quantities can only have n +1 different values, answering to μ = 0, √∙=∙lj μ = l, μ'=0; μ = l, μ' = 1, 2, 3, . . . n- 1. So they will be determined by an algebraic equation of the degree rø+l.
In conclusion, Jacobi mentions researches of his own, which led him to the conclusion that if a modulus κ can be transformed into another λ, they are connected by an algebraic equation of the degree n + l, if the transformation be of the order of the number n, supposed prime. These symmetric equations are, for π=3 and 5, u i - -τ 4 ±2wz>(l - u 2 v 2 ) = 0, u β - ν β + 5u⅜ 2 (u 2 - ν 2 ) ±4ωr(l - u i υ i ) = 0, putting u≈ l jκ, υ= ^∕λ.
These equations he names modular equations, and notices as remarkable that they have their simplest forms when expressed in the fourth roots of the moduli. He also gives the differential equation of the third degree which all these algebraic equations satisfy, viz.,
3(iZκ 2 d 2 λ 2 - dλ 2 d 2 κ 2 ) - 2dκdλ,(dκd 3 ∖ - d ∖ d 3 κ )
÷-O"-m∙∙
Moreover, in some cases the same modulus reappears, and the transformation becomes multiplication. This takes place in all cases when n is the sum of two squares, τι=α 2 + 4δ 2 , κ being V⅜, and the elliptic function becomes multiplied by α±2W. Similarly with all moduli which are connected by any scale with κ ≈ ∖∕⅜, a kind of multiplication not having an analogue in circular arcs.
236. In answer to a request of Legendre that he would furnish him with the clue to his discoveries, Jacobi wrote, April 12, 1828:—“Having found (March 1827) the equation
M = V ifø u ⅛ ’
I recognized that for any number, n, transformation was ^determinate problem of algebraic analysis, the number of arbitrary constants being always equal to that of conditions. By indeterminate coefficients I formed the transformations for the numbers 3 and 5. The biquadratic equation to which the former led me having nearly the same form as that which serves for trisection, I began to suspect some relation. Fortunately I happened to remark in these two cases the other transformation complementary to multiplication. At this stage I wrote my first letter to Μ. Schumacher, the method being general and verified by examples. Subsequently, examining more closely the two substitutions z = α & +ί> & y =
j l + cy 2 j Υ + cx 2 under the form presented in my first letter, I saw that when we put ≈=sin am— , z must vanish, and, as in the said form — was positive, I thence concluded that y must vanish also. In this manner I found by induction the resolution into factors, which being confirmed by examples, I gave the general theorem in my second letter. Having remarked the equation sin am (iξ, κ)≈i tan am(ξ, κ), I next drew from it the transformation from √ to λ'. I had then two different transformations, one from κ to a smaller modulus λ, the other from κ' to a larger one λ'. Thence I conjectured that exchanging inter se κ' and λ, κ and λ', the analytic expression of the complementary transformation would be got. The demonstrations were found only subsequently.”
237. Equally interesting is Legendre’s reply (June 16, 1828):— “As to what you told me of the train of ideas which led you to your beautiful discoveries on elliptic functions, I see that we have both run some risks,—you in announcing discoveries not yet invested with the seal of a rigorous demonstration, and I in publicly and unrestrictedly giving them my full and entire approbation. We have neither of us to repent of what we have done. . . . I saw very clearly that results such as those you had obtained could be no effect of chance or of a faulty induction, but only of a profound theory based upon the nature of things.”
238. Of Gauss’s investigations in this branch of mathematics Jacobi makes mention in his first letter to Legendre (August 5, 1827). These researches “are not the only ones which have been undertaken in Germanyin the same subject. Μ. Gauss having heard of them let me know that he had developed as far back as 1808 the cases of division into 3, 5, and 7 parts, and found at the same time the new scales of moduli referring to them.” Again, April 12, 1828:—“As to Μ. Gauss, he has not yet published anything in elliptic functions, but it is certain he has made beautiful discoveries. If he has been anticipated and perhaps surpassed, it is a penalty due to the veil of mystery he has spread over his works. I am not personally acquainted with him, as I studied philology at Berlin, where there are no distinguished geometers.” Legendre, however, cannot believe that discoveries of such reach can be left unpublished, as was actually the case with Gauss. “If Μ. Gauss,” Legendre writes to Jacobi, April 14,1828, “had fallen upon such discoveries, which in my eyes surpass all hitherto done in analysis, most assuredly he would have lost no time in publishing them.”
239. Simultaneously with the announcements of Jacobi just mentioned there appeared in September 1827, in Grelle, the first part of Abel’s “Recherches sur les fonctions elliptiques,” and accompanying the second part (Feb. 12, 1828) a statement that, “having finished the preceding memoir on elliptic functions, a note on the same functions by Mr C. G. J. Jacobi, inserted in No. 123 of Μ. Schumacher’s Nachrichten, has reached me. Μ. Jacobi gives the following theorem. . . . This elegant theorem, which Μ. Jacobi gives without demonstration, is contained as a particular case in formula 227 of the foregoing memoir (which is the formula of of § 246 infra), and is fundamentally the same as that of formula 270.” This he proceeds to show.
240. The “Recherches” present a great and complete theory of elliptic transcendents. Starting with the inverse function ≠(α) as that determined by α=∕ -.--- and sin 0=<⅛α=or, which by J √l-c 2 sin 2 β ψ ’ j . f^ dx d()N ∖ - sin 2 β = dφa=dx gives a =∕ z , ιι -, —„ , Abel noticed
./0 V(1 - ic 2 )(l - c¾ 2 )
that the formulae become simpler by supposing c 2 negative = - e 2 , and for symmetry writes 1 —c¾ 2 instead of 1 -x 2 , so that the function φα —x will be given by the equation V dx
“T∕ o √(l-c⅛ 2 )(l + e¾ 2 ) ’
or by φ'a = V(l - c 2 ≠ 2 a)(l + e 2 φ 2 a), and for brevity two other functions fa = Vl - c 2 ≠ 2 a, Fa = Vl + e 2 ≠ 2 a are introduced.
After establishing the double periodicity, and determining the zero and infinite values of these functions, Abel proceeds to the development of the formulae of multiplication to determine φlna), f{na), F(nα) in rational functions of ≠(α), /(a), F(a).
He next enters on the solution of the more difficult problem of the division of elliptic functions, which is the principal object of the memoir, Abel proves the algebraic expressibility of the func- ,i ° ns φ (≡⅛1) ’ z (δ⅛i) ' r (⅛) as funotion3 0f ψw ∙ f[a), F(α) in the form
≠0S)- 2 -⅛ j λ(Λ + (c 1 +VC,- - D~+>) s + 1 +∙ + (‰ + √⅛ro^) s ⅛ψ in which
—— 1
⅛β-≠(2n + l)β + ^ f - 1 j (λ,+ √A≈-B∙∙w)≈ iτi + (a,+ √Aj-Br +ι )^+... +(⅛, + √<-Bj+ ι p ι i ·
and the quantities C, D are rational functions of φ l (β), while the quantities A, B are similar functions of ≠(2n + l)3. Thus these equations give φ(β) algebraically expressed by φ 1 (β), and then φ i (β) algebraically by ≠(2n + l)β. So, replacing β by—- α , , we get
(α ∖ '
- - I as an algebraic function of φa ; and similarly for f διi +1 J and F.
241. The priority of this beautiful discovery Jacobi ascribes unconditionally to Abel. To Legendre he writes (March 14, 1829):— “You suppose I have found means of expressing algebraically trigonometric functions of the amplitudes you denote by α m , adding that without that my formula would contain coefficients I could not determine. But that is quite impossible in the general case, and is done only for special values of the modulus. My formula, which gives the algebraic expression for sin am u by means of sin am nu, supposes known the section of the complete function. In this manner, for more than a century, the division of art arc of a circle could be solved algebraically, supposing known that of the complete circumference, this latter having been given generally only in these later times by the works of Μ. Gauss....You see then that Μ. Abel has proved this important theorem, as you call it, in his first memoir on elliptic functions, although he has not dealt in it with transformation, and does not appear even to have thought when he wrote that his forrnulae and theorems would find such an application. The transformed modulus, or, which amounts to the same thing, the regulator, being supposed known, it is still necessary to resolve an equation of degree ⅜(ji-1) to arrive at the quantities sin 2 arn(2y>ω), or at the section of the complete function. Thus you had only to solve a quadratic in the case of τι=≈5. Μ. Abel proved that Μ. Gauss’s method applies nearly word for word to the solution of these equations, so that it is only the modular equations that we are unable to solve algebraically.”
242. Starting from the solution of the problem of multiplication and division, ≠(2n+l)3 is exhibited by Abel as the quotient of two double products, the factors of which depend linearly on ≠(β). Thence, putting3 = , and n≈∞, the development of the in verse function ≠(α) is derived in double products and double sums, the factors of which are linear in α; accordingly a unique analytic expression is found for the function heretofore defined only by its properties. The reduction of the double products and double sums to simple products and simple sums, the product development and breaking up into partial fractions of the elliptic functions, follow then without further difficulties.
243. With the publication of the “Recherches” Abel clears at one bound the limits of the investigations of Jacobi hitherto published, though the first part devotes no attention to the problem of transformation of elliptic integrals. Moreover, this work drew from Gauss the remark:—“Μ. Abel has anticipated me at least in a third part. He has just trodden precisely the same path I went along in 1798. And so I am not at all astonished at his arriving, for the most part, at the same results. Besides, as in his deduction he has displayed so much sagacity, penetration, and elegance, I feel myself by it relieved from the publication of my own researches.”
244. The same volume of Grelle contains, besides the first part of of the “Recherches,” indications in the paper “Problems and Theorems” that Abel was at the time in possession, not only of the theory of rational transformation which Jacobi treated, but of the general algebraic transformation, as has been made manifest subsequently in his collected works.
245. Before Jacobi had read the “Recherches” he published a proof of the general theorem of rational transformation in No. 127 of Schumacher’s Nachrichten, December 1827. It is based on enumeration of the constants available, and fixing the conditions in order that the substitution y = H-may satisfy the differential equation "
dy
√(1 - α 2 ∕)(l - a ’ 2 ∕)(l - a"y)(l - a ’ "y) _ dx
~M√(l - ∕3≈)(1 - 0⅛)(1 - 0"ae)(l - 3"⅛) ’
He introduces the unique inverse function which he calls sine of the amplitude, sin am, and gives the value of
(W⅛-y... ∕ 1 -.. « y ∖ sin coam - ∕ ∖ sin coam ∕
- X 2n + l∕ ∖ 2n + l∕
I —— ≡E≡ I. , i I l ...
(1 - ∕c 2 z 2 sin 2 am ∖ . . . ( 1 - κ⅛ 2 sin 2 am —ft ∖
∖ 2n + 1J ∖ 2n+l∕
as satisfying the differential equation (M being constant)
dx m dy
λ∕(l - ar i )(l - κ¾ 2 ) √(1 - y 2 )(l - λ 2 y 2 ) ’
The value of y is derived from this, and Jacobi remarks that this theorem holds generally, but does not embrace all the solutions of the problem.
246. The second part of the “Recherches” was finished by Abel February 12, 1828, and appeared immediately in Grelle. The first problem treated is the algebraic expressibility of the function φ(^ — when certain relations, as for the lemniscate, hold between e and c. The principal application of this is the expression of the function by square roots whenever n is of the form 2 n or 1 + 2 n , the latter being prime. He then proceeds to deal with the general treatment of rational transformation, which he presents in the following form.
If α be ( wt +ft)ω + θft-∕⅜)ari where at least one of the integers 2ra+l ’ s m and μ is prime to 2n + 1, we shall have f dy . Γ dx J [(1 - <∖V)(1 + e 2 M i J [(1 - c⅛ s )(l +t⅛ 2 )]* ’
where v =f x (Φ 2a ~ x 2 )(.Φ‰ - ≈ 2 ) . . (φ 2 na - a?) λ ~
' (1 ⅛e 2 c 2 ≠ 2 α. x 2 )(l + e 2 c 2 <∕> 2 2a . ae 2 ). . . (1 + e 2 c 2 φ 2 na . x 2 ) ’
⅜^4[ ≠(⅞ +a ) ≠(⅜ +2 ∙) ≠(⅜+~)]^ - ⅜4[<⅞÷√<lH∙∙√WΓ-
a=∕(≠a . φ2a . φ3a . . . φna) 2 , f being an indeterminate, so that there only exists a single relation between the quantities c 1 , e 1 , c, e. The section concludes with the words—“To have a complete theory of the transformation of elliptic functions, it would be necessary to know all the transformations possible; now I have succeeded in demonstrating that they are all got by combining that of Μ. Legendre with those contained in the above formula, even when we are looking for the most general relation between any number of elliptic functions. This theorem, the consequences of which embrace nearly the whole theory of elliptic functions, has led me to a very great number of fine properties of them.”
247. The same number of Grelle contained, in an extract from a letter by Jacobi,“Note sur les fonctions elliptiques,” the exhibition of sin am as the quotient of two series Θ and H, or as they were subsequently called the θ and 0 1 functions; also appended to √∩τr ~-* , K'
— by powers of q≈e κ j the expon- 7Γ
ents of which are the squares of the natural numbers; as also the very important development of V/c as quotient of two series proceeding by square powers of q and q⅛— results whose importance was at once accepted by Legendre. Regarding tho result in this paper, that to a given modulus for a prime degree of transformation n there always correspond n +1 other transformed moduli got by i_ J. putting q n , q n , aq n , . . . a n ~ 1 q n for q, where α n =l, Jacobi remarks, ‘ ‘ Thus Μ. Abel will see that imaginary transformations had not escaped me.”
248. Jacobi’s well-known construction for the addition and multiplication of elliptic functions by the arcs determined on a circle by the vertices of an inscribed polygon, whose sides touch other circles coaxal with it (or, as he entitled it, application of elliptic transcendents to a known problem Qf elementary geometry—to find the relation between the distance of the centres and the radii of two circles, one inscribed in, and the other circumscribed to, an irregular polygon), is of about the same date, April 1, 1828.
Immediately afterwards, Jacobi, still ignorant of the second part of Abel’s “Recherches,” communicates to Legendre (April 12, 1828), the forms of development detailed in the work we have just spoken of for the sin am, for the modulus of the integral, and for the period K, and notices that these formulae will not be without interest for the celebrated geometers who are engaged with the motion of heat,—numerators and denominators of the fractions by which the trigonometric functions of the amplitude have been expressed being often met with in that question.
249. Abel sought to generalize the problem of transformation, in the publication of which he was anticipated by Jacobi.
“We may regard this theory” he says (May 27, 1828), “from a much more general point of view, proposing as a problem of indeterminate analysis to find all possible transformations of an elliptic function which can be effected in a certain manner. I have attained the complete resolution of a great number of problems of this kind,—among them the following:—To find all possible cases in which we can satisfy the differential equation dy dx
√(1 - c 1 2 y 2 )(l - ¾V) = ±a V(l - c¾≈ 2 )(l →⅛0 · · · · (1) by putting for y an algebraic function of x, rational or irrational. This problem may be reduced to the case that y is rational. For we can show that, if (1) holds for an irrational value of y, we can always deduce from it another of the same form in which y is rational by suitably changing the coefficient a,— the quantities c 1 , e 1 , c, e remaining the same. The first method which presents itself for resolving this problem when y is rational is that of undetermined coefficients. But this is a very fatiguing process. The following, I believe, deserves the attention of geometers, leading as it does to a complete solution in the simplest manner.”
The theorem of the reducibility of the general problem of transformation to the rational is, however, stated without proof in this paper, but the problem of rational transformation, based on considerations of periods for the original and transformed elliptic function, is strictly treated. It is shown to resolve into simpler analogous problems whenever the number characteristic of the transformation is a compound one, and the equation of transformation itself is stated to be algebraically soluble. Lastly, Abel enters more closely into the case of equality of the transformed moduli of the integrals (viz., e i =e, c 1 = c), which has subsequently constituted the theory of the complex multiplication of elliptic functions. The multiplier a of the transformation is foιuιd in the necessary form μ' + ∖ J - μ, where μ' and μ signify two rational numbers, of which the latter must be essentially positive; and Abel adds—“If we attribute to a such a value, we can find an infinity of different values of e and c which render the problem possible. All these values are expressible by radicals:” Regarding the subject of this paper, Jacobi remarks to Legendre (June 14, 1829), “Abel’s principal merit in the theory of transformation consists in his demonstration that our formulae embrace all possible algebraic substitutions, and this gives a high degree of perfection to this theory.”
250. In the “Suite des notices sur les fonctions elliptiques,” dated July 21, 1828 {Grelle, vol. iii,), Jacobi introduces his functions Θ and H as independent fundamental functions on which to base the theory of elliptic transcendents; a conception to which also Abel was simultaneously led, and which he gave utterance to in writing to Legendre, Nov. 25, 1828:—“The theory of elliptic functions has led me to consider two new functions which enjoy several remarkable properties.” Abel desired, quite in analogy with Jacobi’s principle, to treat of the properties of these new transcendents apart from the inverse function of the elliptic integral, but the priority of publication of this discovery is Jacobi’s, since the completion of the second part of the “Précis d’une théorie des fonctions elliptiques,” which was to contain all these investigations, was inteιτupted by Abel’s unexpected death.
The above-mentioned work by Jacobi next contains the theorems expressing elliptic integrals of second and third kinds by θ functions. Regarding the formula of reduction of the integral of the third kind by aid of the θ functions, Jacobi remarks a characteristic property to Legendre (September 9, 1828):—“Moreover it shows that elliptic functions of the third kind into which three variables enter reduce to other transcendents which contain only two,”—a discovery which Legendre was greatly interested in, though he found difficulties in the distinction of real and imaginary parameters, expecting that the introduction of an imaginary parameter involved three independent .quantities in the integral, and so there should be four kinds of elliptic functions instead of three.
Jacobi, however, repeats the above assertion in his demonstration of the relation Π(u,α) = ιtZ(α) + ⅛ log θ( ιt ~ . ffl ) (C⅛¾Z⅛,iv.):—“This
Θ(l6 t , Ce) latter formula shows that elliptic functions of the third kind which depend on three elements can be reduced to other transcendents which contain only two.” Later, Jacobi wrote to Legendre (May 23, 1829):—“As regards elliptic integrals of the third kind with circular parameter, they do not admit of a reduction analogous to that of the logarithmic kind. In a general analytical sense not distinguishing between real and imaginary values, a formula embraces all cases; but in applying to numerical calculation cases must be distinguished.” And again, “as to the numerical calculation of elliptic integrals of the third kind with circular parameter, I do not think you should too much regret the inconvenience that they cannot be reduced to tables of double entry.”
251. The collected statement of his investigations, published by Jacobi as his Fundamenta nova Thcoriae Functiornιm Ellipticarum in 1829, contains two main divisions, the first on the transformation of elliptic functions, the second on their evolution. We have already indicated many of the principles contained in this work, the most important of all being that of the double periodicity of these functions. As Jacobi says (Works, i. p. 262)—“Elliptic functions differ essentially from ordinary transcendents. They have, so to say, an absolute manner of existence. Their principal character is to embrace all there is in analysis that is periodical. In fact, trigonometric functions having a real period, exponentials an imaginary period, elliptic functions embrace the two cases, since we have at the same time sin am (u + 4K) = sin am«, sin am (w + 27K') = sin am u. Moreover it is easily demonstrated that an analytic function cannot have more than two periods, one real and the other imaginary, or both imaginary (complex) if the modulus k be so too. The quotient-^ of the periods of a proposed function determines the modulus of the elliptic functions by which it must be expressed by means of the
(the expansions for which in terms of q follow by § 258). Perhaps, it will be convenient to introduce this quotient into analysis as modulus in p±ace of k. ’ ' On these principles Jacobi subsequently founded a theory of hyperelliptic functions.
252. Jacobi’s first evolution of elliptic functions is into infinite products, and is derived from the transformation from λ to k', which is expressed by sin arn(««, k)
= ∕∆-sinam X sin am (=?£-+ — ) sinam + —)∙∙∙
V k · μ ∖M n J \Μ η J sin am + , and equivalent forms, by writing in the equations of transformation, for u, and allowing n to take an infinitely great value. When this is done, am , becomes am I —. χ — ,=a:, and he arrives at the equations
\«M J 2k
2Kx 2⅛d . (1 - 2j 2 cos 2α + 2 4 )(l - 2q i cos 2x+i 8 )...
sin am—= sin » (1 „ 2 g cos 2z+j i )(l - 2 2 3 cos 2x+q*)... ’
2Kz , 1 ∕P (l + 2î 2 cos2ae + ÿ 4 )(l + 2i 4 cos2^+2 8 )... /AX cos am—=2 7 i χ ∕ r cos _ 2ycoaaι . +s . χi _ 2 g* cos 2 α Ι√‰ (A) '
2K.r_ ∕i√ ( 1 ÷ ¾ cos 2z ÷g 8 )( 1 + ¾ 3 cos 2aι + g β ): Aarn— = V A; ( 1 _ 2îco9 2 â : + 2 2)(1-2î 3 cos22?+s 6 )... , from which are easily derived such series as
27* K . 2Kz 4 √g^sin x 4 √g 3 ^sin 3¾ 4√g 5 sin 5a?, &c> - a nam-≡--+ + 1 ^ 8
_ 4 sta U √<0÷ 8)., + v ⅝ + ⅞-. + 4 0 ,1.
( 1 - 2q cos 2x + q' 2 1 - 2q 3 cos 2x + q 6 j The series for powers of these functions are then investigated; and it is found, e.g., that the square may be written ^B^⅛J l sin 2 am 4K(K-E 1 ) (2g cos 2x ι 4g 2 cos 4x ι 6q 9 cos 6x )
π 2 ( 1 — q 2 1 - g* 1 - g β ^*^ )*
253. This enables the second kind of elliptic integral to be evolved in a series. The form introduced being called Z is related to Legendre’s E by the equations
2Kz . F 1 E(φ)-E 1 F(≠)
— = «, ≠ = amu, Z(u)= i22 ,
7Γ Γ
aud the expansion is found
2K_ ∕2Kx∖ 2Ka√2K 2E¼ ∕2fcK∖ 2 Γ x . o 2Kx, T z (-rv(v-τ)-(-M ≡-≡^< to ∙ (B) . j q sin 2z ι q 2 sin 4x ι q 3 sin 6x ∣
( 1-2“ + l-2 4 + T-2 β > '
254. Before proceeding to the serial development of the third kind of integrals, the theorems concerning their reduction to depend on functions containing only two variables are given. It is shown first assuming two angles <r, δ, such that F(≠) + F(α) = F(σ) and F(≠) - F(α) = ι F(δ) that
∕∙Φ k 2 sin a cos a ∆a . sin 2 <ft dφ jo {1 - Ae 2 sin 2 α . sin 2 ≠ } Δ(≠)
. ri ,< λ -Z'⅛⅛...∕⅛ . . (C) 1 Vo A>(≠) √ 0 Δ(≠)
so that the third kind of elliptic integrals, which involves three elements, the modulus k, the amplitude φ, and the parameter α (the quantity-⅛ 2 sin 2 a is what Legendre called n the parameter, § 206) is reduced to integrals of the first and second kind, and to the new transcendent ∕ , each of these depending only on two elements.
This new transcendent we see from the above equation, by letting F(α 2 ) = 2F(α), and so σ=a 2 , δ=0, for φ=a, satisfies the equation ∕ α ⅛ 2 sin α cos a ∆a . sin 2 ≠<7φ Γ a " l ↑L(< ∣ >)dφ
(l-* 2 sin 2 asin 2 ≠)∆(≠) = F W E (a )-y 0 7∆ (≠Γ ’
that is to say, for the new transcendent we may substitute the definite integral of the third kind in which the amplitude is equal to the parameter; another statement of the reducibility to functions depending on two elements only.
The above equation (C) may be transformed by the identity derived from the formulae of § 207, sin 2 σ-pj ιa g⅝~ 4 sinα eos « ∆a sin ⅜ cos ⅜ Δ ≠
(1 -⅛⅛in 2 a sin 2 ≠) β which gives, on introducing am «for φ, am re for α, and consequently am (« + α) for σ and am (« - α) for δ, and integrating, ∕ u
<7w{sin 2 am(w + α) - sin s am(u - re)}
_ 2 sin am a cos am re Δ am re. sin 2 am «
1 - ⅛ 2 sin 2 am a. sin 2 am«
255. Jacobi accordingly accepts as canonical for an integral of the third kind the form written above. He defines it by
, , _ ∕" t % 2 sin am α. cos am re. Δ am re. sin 2 am «. du
11(14. CL)—∕ rs—;—x —5— y— ·
√θ 1 - κr suram α. suram u PZ(v)du
Again, denoting by Θ(u) the expression Θ(w)≡≈0(O)e 0 , the integration of the series for Z(u) (§ 253) gives
2K ∕' x ∕2Kx ∖ , lqcos2x q 2 cos4x , q 9 cos&x ι . ) 1 z (-r-- 2 !τ^ + 2<w ij j+⅛r^+*'>∙i+<≡t =log ft* -¾CQ8¾e÷ g 8 )(¾ -2g 8 cos 2x + 2 β )(l ~ 22 g cos 2¾ + 2 10 ) - -)
' {(l-g)(l-√)α-V) · · ∙}* i 5 whence e √2KαA
∖ ιr ) (1 - 2g coβ 2a?+2 2 )(l - 22 3 cos 2x +g 6 ). . .
θ(°) {(l-g)(l-g 3 )(l-g β ). . ∙} 2
256. This is the first introduction in the Fundamenta of these functions, which have been called theta functions from the original notation adopted for them by Jacobi, and by many writers have ceen named after him Jacobian functions.
The connexion of the integral of the third kind with these functions follows at once from g 254. In fact, calling - ©'(«), it is du
∏(w, re) = «Ζ(α) + ⅜ log = « ¾7⅛ + ⅛ log
’ , o Θ(w + re) ©(re) 2 0 Θ(u + a)
But, since © is an even function in ©(«)·=©( - «), whence ∏(a, «) - aZ(«) + ⅛ log θ⅛7-¾ .
©(«+a)
Hence, subtracting,
Π(«, a)-∏(⅛, «) = wZ(a)-aZ(«), which is in this notation the theorem that an integral of the third kind can always be reduced to another in which its parameter ana amplitude are interchanged, as was noticed by Legendre (§ 216).
The development of n(«) in a series is found by aid of the series for stn 3 am « and of the last equation in § 254. It is π (2Rr 2KA) 2K¾ z (2KA)
o I 2 s i n 2A sin 2ae ι 2 2 sin 4 A sin 4x <7 3 sin sin 6ae 1 s (
' 2 I W + ~2(l-2 4 ') + ' 3(1- 2«“+ I ·
257. Returning from the integrals to the elliptic functions, the expressions in infinite products (A, § 252) are resumed. The occurrence of the function Θ is apparent in each of the denominators. Introducing the definition of a function H, jj∕ 2Ka,∙ ∖
∖ π J _ 2(fi sin ¾(1 - ¾2 a c ° 8 2a?+2 4 )(1 - 22 4 cos 2x + 2 8 ) · · ·
•W = {(1 -iXl-i3)(1 -2 5 )...p and replacing2-i⅛ by «, it is easily seen that the expressions are 7Γ
sinam«- 1 H(ît) cosam«= /^H(« + K) √⅛ ©(«) V k ©(«)
∆amu=√⅛'θ^±2S∖
Again, it is easily seen that ©(« + 2K) = ©(«) and H(« + 2K) = - H(«). Also as by substituting iu for « (§ 235) we get iZ(fu, k)≈Z(u, fc ’) +-JpL _ tan am(«, ⅛')∆aιn(re, k ’ );
2K.K.
whence, integrating,
.πw 2 m 4KF cos am(W) ,
®(0, k) ’ Θ(0, k ’ ) ’ ιr(K ’ -iu) it follows that Θ(« + 2ιΚ')— -e κ ©(«) ;
π(K ’ -2iκ)
as also that ©(«+ι - K') =ie ©(«). V⅛ sin am «.
π(K'-2⅛)
whence =ie 4K H(«); and by successively replacing « by « + iK' it is hence seen that ©(«) and H(w) have one real period 4K common with the elliptic ∙>rU 2 τru a functions, and that e4E· K '©(«) and e4KK 7 H(w) have another imaginary period 4iK' in common with them.
258. The expansion of the θ and H functions in series of cosines and sines of multiple arcs by the method of indeterminate coefficients determines θ (~π~)≈ 1 - 22 cos 2» + 22 4 cos 4x - 2q 9 cos 6x + 22 lβ cos Sx -,
∕ 9!Tζy χ
)~22⅛ sin X-2q^ sin 3sj+22^ sin 5a:- 2q^ sin 7∙τ+; and hence a new development of elliptic functions as well as of the integrals arises.
The developments of the numerators of the cos am and Δ am may be written down from the above—in the notation subsequently used by Jacobi—
θx=^l - 22 cos 2a:+22 4 cos 4x- 2q 9 cos6x+ . . . θ 1 x≈2qi sin a:— 2q^ sin 3x+2i^sin 5x - . . . θ 2 x=2qi cosx+2q^ cos 3xk2q^ cos 5x+ . . .
θ 3 x≈l + 2q cos2aj÷22 4 cos 4x + 2q 9 cosβx+ . . . as, for instance, in his lectures, in which, without any presupposition from the theory of elliptic transcendents, he established the relations which these series fulfil, and from them a theorem of addition for the quotients of the series, and from this the differential formulae which lead immediately to the elliptic integrals. All these formulae^ consist of series of exponential quantities, extending in both directions to infinity, in which the ordering element in the exponent rises to the second degree. Their general form may therefore be written se ay2 + 2 b v + c , where v takes all positive and negative integer values.
259. The Fundamenta Nova appeared almost at the date of the death of Abel. Of Abel’s works, besides those which we have mentioned, the chief is the unfinished “Precis d’une théorie des fonctions elliptiques,” which appeared in Grelle, iv., in 1829. “The whole of my researches will form a work of some extent which I cannot yet publish, therefore I give here a , Precis ’ of the method I followed, and its general results.” The fragment of this work which has been published deals only with the integrals.
260. The consideration of the indeterminateness of the integral f x dx U= J Q √(l-x 2 )(l-⅛ ’
which gives rise to periodicity in the inverse function x = sin am u, has led to the consideration of the whole subject from a new point of view. The introduction of the complex variable into analysis by Cauchy in his Mémoire sur les intégrales définies prises entre des limites imaginaires (1825), and by Gauss in the second part of his Theoria Residuorum Biquadraticorum (1831), has been followed by the works of Puiseux (“Recherches suries fonctions algébriques,” Lionville, xv. 1850), of Riemann {Inaugural Dissertation, 1851, and “Theorie der Abelschen Functionen,” Grelle, 1857), and of Weierstrass (“Theorie der Abelschen Functionen,” Crβlle, 1856)—which develop the subject in this more extended field, perfecting the conception which the term function covers in analysis, and pointing out the essential distinctions in the different modes of dependence of two quantities,—such distinctions, for instance, as when a function is defined by a differential equation, whether it is one-valued or not, and, if it be, whether it is integer or fractional.
261. In close connexion with this is another department to which the theory of transcendents has with great success been applied,—the investigation of the geometrical properties of curves. The points on a curve are expressed as functions of a parameter, and on the nature of these functions the nature of the curve depends,—the “deficiency” or “Geschlecht” of the curve (see Curve, vol vi., p. 725) determines the nature of the function, and any curve into which another can be rationally transformed depends on the same function.
We shall conclude with a brief application to the case of elliptic functions and plane curves of the third degree. It is well known that the equation of any non-singular cubic can be reduced to the form yz i =x{x-yfx-B 1 y) , where y = 0 is the tangent at the point of inflexion in which the curve meets ae=0, and ae=0, x=y, x=k 3 y are the tangents from that point to the curve, their points of contact lying on z = 0.
This equation is satisfied identically by assuming the equations pz=sin am u, ρy=sin 3 amu, p~=∆arn u cos amw, which determine any point on the curve by a parameter u. To each value of u corresponds a perfectly definite point of the curve. But on the other hand, to any point of the curve corresponds an infinite number of values of the argument all related to one of them, u,— differing from it only by a multiple sum of the periods.
The occurrence of the elliptic integral u here in this normal form results from the coordinates chosen; but, whatever they be, we see that the points of the curve can be expressed by a parameter depending on no higher irrationality than that we have introduced. When the cubic has a double point, the coordinates of any point on the curve can be expressed by a parameter without introducing any irrationality.
262. To investigate the intersections of the cubic with a right line we proceed to derive in a simple manner a slight extension of Euler’s integral (§ 207). Written in Jacobi’s notation it is cosam(u 1 + w 2 ) = cos amω 1 cosam w 2 - sinamu 1 sinamw 2 ∆am(w 1 + w 2 ), which is easily thrown into the form k' 2 + k 3 cos am u l cos am ∙u 2 cos am(w 1 + w 2 ) = ∆am u l ∆am u 2 ∆am(w 1 + u 2 ) .
This may be extended to three arguments as follows. Denoting sin am u τ briefly by s r , also cos am u r by c r , ∆am u r by ∆ r , tan am u r by t r , and cot am u r by ct r , the formula may be written kl' 2 +k 3 c 1 c 2 c(u 1 + w 2 ) = ∆ 1 λ. 2 ∆( w 1 + w 2 ); putting for u 2 , u 2 + u 3 , and expressing, by § 207, c(w 2 + w 3 ) and ∆(w 2 + t⅛) by functions of one argument, we get kl 2 + Zr 2 c 1 c 2 c 3 c(ιt 1 + u 2 + w 3 ) - ∆ 1 ∆ 2 ∆ 3 ∆(u 1 + u 2 + m 3 ) ≈⅛¾ 2 s 3 (⅛⅝ 2 s 3 + c 1 ∆ 2 ∆ 3 c(ti 1 +w 2 + tt 3 ) - ∆ 1 c 2 c 3 ∆(t¾ + w 2 + u 3 )} .
Now the former expression is symmetrical; denoting it by k 2 s 1 s 2 s 3 θ, yve can determine θ as follows. Writing for brevity c(u 1 + u 2 + u 3 ) = C, and ∆(u 1 + u 2 + w 3 ) = ∆ , the equation is ⅛' 2 s 2 s 3 + c 1 Δ 2 Δ 3 C - ∆ 1 c 2 c 3 ∆ - 0s 1 = 0 .
Hence writing down the three equations, which must hold from symmetry,
0=V⅛ 2 s 3 + c 1 Δ 2 Δ 3 C - ∆ 1 c 2 c 3 ∆ - 0.9 1 , 0 = ⅛'¾ 3 s 1 + c 2 Δ 3 Δ 1 C - ∆ 2 c 3 c 1 ∆ - θs 2 , 0= B 2 s 1 s 2 + c 3 Δ 1 Δ 2 C - ∆ 3 c 1 c 2 ∆ - θs 3 , we obtain C and Δ as quotients of determiûants of single arguments. For Δ we get [table]
and for C [table]
But in this, increasing each argument by iK , , since then we get, for s r , -J > for ⅛, ~r~~ > for Δ,, -ict r , for t r , — , and thus ks r ks r ∆ r ’ f∆
for C, , where S stands for sin am(tt 1 + w 2 + u 3 ), this formula gives [table]
whence [table]
and the value for θ thus found gives V 2 + ⅛ 2 c 1 c 2 c 3 C(w 1 + w 2 + w 3 ) - ∆ 1 ∆ 2 ∆ 3 ∆(w 1 + u 2 + u 3 )
+ A7 2 A∕¾χS 2 s 3 S(Wχ 4- u 2 ∙f∙ w 3 ) = 0 ·
263. The formula thus obtained for sin am(¾+M s +¾) vanishes when w 1 + w 2 + t⅛ = 0, or differs from 0 only by an integer combination of the periods. But the determinant [table]
vanishes if its constituents be the coordinates of three collinear points. But these are, as we have just seen, the coordinates of three points on the cubic yz 2 ≈x{x-y){x-k 2 y), § 261.
This result may therefore be stated thus :— If the points of a cubic be expressed as elliptic functions of a parameter, then for the intersections with a right line the sum of the arguments differs from zero only by some integer combination of the periods.
This enables us to solve many problems. For instance, the arguments of the points of contact of the four tangents which can be drawn to the curve from a point u on it are u u + ω u + ω ’ u+ω+ ω' ~^2 , 2~ ’ 2~ ’ 2
Conversely the tangential point u of a given point v of the curve is determined by u≡ - 2v (mod co, ω').
The problem of determination of points of inflexion when one point of inflexion is known is identical with the problem of the special trisection of elliptic functions, i,e,, of the determination of the values for u=f ,ω + l l~ to the same moduli.
3
Bibliography — In addition to the works on elliptic functions and the higher transcendents already named, there have recently appeared as independent works, besides innumerable memoirs in the various mathematical periodicals—· Briot and Bouquet, Théorie des fonctions elliptiques, 2d ed., 1875; Briot, Théorie des fonctions abéliennes, 1879; Booth, Theory of Elliptic Integrals, 1851; Casorati, Teorica delle funzioni di variabili complesse, 1868; Cayley, Elementary Treatise on Elliptic Functions, 1876; Clebsch, Geometrie, 1876; Clebsch and Gordan, Theorie der Abelschen Functionen, 1866; Durège, Elemente der Theorie der Functionen, 1873; Id., Theorie der ellipt. Fund., 1878; Eisenstein, “Beiträge,” collected in his Math. Abhandlungen, 1847; Ellis, “Report on Recent Progress of Analysis,” Brit. Ass. Reports, 1846; Enneper, EU. Fund. Theorie und Geschichte, 1876; Königsberger, Transformation, <Scc., der EU. Functionen, 1868; Id., Theorie d. EU. Fund., 1874; Id., Theorie d. hyperelliptischen Integrale, 1878; Id., Geschichte der EU. Transcendenten, 1879 (to the last two authors we are mainly indebted for the historical details of our subject); Lipschitz, Differential und Integralrechnung, 1880; Neumann, Ueber Riemann’s Theorie d. Abelschen Integrale, 1865; Roberts, On Addition of Elliptic and Hyperelliptic Integrals, 1871; Russell, “Report on Recent Progress in Elliptic and Hvperelliptic Functions,” Brit. Ass. Reports (1869, p. 334; 1870, p. 102; 1872, p. 335, 1873, p. 307); Schellbach, Ell. Integralen und Theta Functionen, 1864; Schlömilch, Compendium der Höheren Analysis, 2d ed., 1874; Sohncke, “Elliptische Functionen,” in Erseh and Gruber’s Encyclopädie-, Thomae, Abriss einer Theorie der complexen Functionen, 1873; Id., Sammlung von Formeln, 1876; Verhulst, Traité élémentaire des fonctions elliptiques, 1841; Weber, Theorie d. Abelschen Functionen vom Geschlecht 3,1876; weyr, Theorie d. EU. Fund., 1876; and of the highest historical interest is the publication by Borchardt of the correspondence between Legendre and Jacobi in vol. lxxx. of CreUe’s Journal, 1875, reproduced in vol. 1. of the collected works of Jacobi, 1881.
