If we pass over the speculations of the earliest evolutionists—De Maillet, Maupertuis, Robinet, Bonnet, &c.— as too vague for rapid summary, and note that Linnaeus was not quite a consistent creationist, since he admitted that many species may be simply fertile hybrids, the history of definite speculation as to the factors of variation may be fairly said to begin with Buffon, whose distinct though covert suggestion of the doctrine of descent with modification was supported by emphatic insistence upon the importance of external conditions. He endeavoured to show how “natures, instincts, and most inward qualities” are modified through bodily habits, how new functions seem to arise in response to new conditions, and how changes in climate, food, and other conditions of life bring about direct modifications in organisms exposed to them. Kant, on the other hand, viewed the evolution of species as related essentially to the mechanical laws of the organism itself, although in his latest writings he allowed for the influence of geographical distribution, food, &c., noted the importance of selection in artificial breeding, and even hinted at the notion of struggle for existence, which was soon afterwards more clearly emphasized by Herder (Ideen z. Phil. d. Geschichte, 1790) in a striking passage, which may be briefly quoted in free translation: “All is in struggle, each one for himself. . . . Space is too small for the number. . . . Each genus looks after itself as if it were the only one. . . . In this way the whole was preserved. . . . Thus were forces cradled, and limbs counted, and tendencies determined, and the earth came to bear what it could.”

Erasmus Darwin {Zoonomia, 1794) believed that the organism has the faculty of improving by its own inherent activity, that it has the power of attaining new parts attended with new propensities. Yet the strongest and most active animals are those which propagate and hand on improvements. Transformations, too, may be produced in part by the exertions of the organism in consequence of its desires for food, security, and reproduction. Changed conditions, such as climate, have an indirect influence in changing desires, and thus actions, and so finally structure; and they may also operate directly. Treviranus (1802-1831) assumed an indefinite variability of the organism, with considerable power of adaptation to surroundings, and even anticipated much recent speculation in his suggestion of a possible factor of modification in the union of sexual elements. The well-known theory of Lamarck (1801-1809) laid special emphasis on function and environment; for, though the sense of need in association with suitable environment calls out a succession of efforts, and so originates incipient structural modifications, it is to increased functioning that the increase of these modifications must be ascribed, while similarly disuse explains degeneration. Changed conditions produce new wants, nutritive and reproductive; hence changes in climate, or the like, change the organism by changing its habits. Rapid increase is checked by other organisms: the strongest and best-armed for attack devour the weaker, and the less perfect genera are kept down by the more perfect. The less definite view of Goethe included, besides recognition of the conservative or centripetal force of heredity, that of a progressive or centrifugal tendency to adaptation to environment. Oken (1809) similarly regarded all progress from his primeval “Urschleim” as having been in terms of its interaction with the external conditions of life. In 1813 Wells made his now well-known suggestion of the importance of natural selection in determining the varieties of the human species; and in 1831 Patrick Matthew published his much more developed, yet equally disregarded, statement of the same doctrine in its more general applications. In 1828-30 Geoffroy St-Hilaire, afterwards ably succeeded by his son Isidore, denied indefinite variation, regarding function as of secondary importance, and laying special stress on the direct influence of the environment: for instance, it was not so much the effort to fly as the (supposed) diminished proportion of carbonic acid in the atmosphere which determined the evolution of birds from saurians. The veteran geographer Von Buch naturally inclined to emphasize the influence of geographical isolation (locality, climate, soil, food, &c.), and laid stress on the restriction of the area of possible sexual union as bearing upon the origin of varieties. The embryologist Von Baer (1834) dwelt especially upon the organismal nature of variation, on the unfolding, as it were, of new structures; and Schleiden (1838) and other naturalists more or less distinctly advanced similar opinions. In 1844 appeared the Vestiges of Creation, which in its later editions (1853) formulated an hypothesis of progress

(1) by rhythmic impulse through grades of organization,

(2) by another impulse tending to modify organic structures in accordance with external circumstances. In 1852 Naudin argued for the formation of new species in nature in a similar way to that of varieties under cultivation, further attaching great importance to an assumed “principle of finality,” apparently a kind of organismal fate. Herbert Spencer, whose weighty arguments for evolution date from 1852, laid special stress upon the modifying influences of environment, this involving changes of function, and so ultimately of structure. Finally, in 1853 Victor Carus argued for a progressive adaptation to changing external conditions. An undefined hypothesis of internal modifiability appears also to have floated before the mind of Owen.

In this succession of variational theories we recognize the repeated general insistence upon every separate factor in the problem,—organism, function, and environment,— the successive authors, in fact, almost falling into as many schools. Yet these are singularly unprogressive: the same hypothesis is stated again and again, but always fails to carry conviction; many employ the merest abstractional explanations, in terms of what is simply the old pre-physiological “vital force” (see Biology), or attempt to cut the knot by the more or less open introduction of a deus ex machina. Even the more definite and scientific theories of Erasmus Darwin, Lamarck, Geoffroy, and Spencer fail adequately to establish in a sufficient number of concrete cases the supposed potency of environment or function, much less of internal or volitional moods, in determining extensive changes; moreover, they are constantly confronted by cases in which any one set of changes seems to take place independently of the other. Hence, despite the deep and unanimous conviction of so many biological thinkers as to the origin of species by evolution, the essential problem remained unsolved. For, until some explanation, rational and concrete, detailed yet universal, of the origin of species should be reached, the cautious naturalist could not feel justified in accepting as a scientific certainty a doctrine which rested so largely upon undemonstrable grounds. Any really new attempt at establishing the doctrine of evolution had thus, not simply to reorganize and strengthen the empirical evidence, by marshalling beside the generalizations of morphological unity of type the facts of more strictly historic bearing,— distributional, palaeontological, and embryological,—but also to meet this deeper theoretic want by showing not only that evolution has taken place but how. Hence the importance and rapid acceptance of the Origin of Species (1859), for which the time was fully ripe. For it is evident from the preceding historical outline that the view still frequently promulgated of the exclusive and, as it were, catastrophic importance of this work in the establishment of the doctrine of evolution must be replaced by a more uniformitarian and, so to speak, less creationist view. It should not seriously be supposed that the theory of descent which had been held in one form or another by the leading biological thinkers of that and the two preceding generations, with almost the individual exception of Cuvier, had been decisively abandoned by men of science, or that the public which had purchased ten editions of the Vestiges of Creation in as many years, and which took up the first edition of the Origin of Species on the day of publication, could be quite unprepared for the acceptance of the major thesis of both works. Yet that a lull had taken place is also unquestionable, for the failure of wave after wave of speculative effort had produced a widespread feeling of discouragement, and a tendency to concentration upon more concrete and smaller (for the most part Linnaean or Cuvierean) problems alone, since of these the solution was comparatively sure. The dramatic coincidence between Darwin’s and Wallace’s conclusions soon helped to gain immediate attention to the new theory; and this was thoroughly developed and retained by the prompt appearance of the first and second editions of the Origin of Species. Without entering upon a review of this classic treatise, much less a retrospect of the controversy it provoked, it is evident that it was upon almost all sides so much stronger than preceding attempts, its central doctrine so lucid and obvious a statement of the everyday facts of life, its mode of presentation so temperate yet so forcible, its self-criticism so frank and unconcealed, that its marvellous success was owing far more to intrinsic merit than favourable circumstances. The required re-statement of the evidence for the historic fact of evolution was at last learnedly and forcibly done, and the salient difficulties in its way, like that of the imperfection of the geological record, admirably met; yet this portion of the work sinks into comparative ineffectiveness beside the long-sought modal explanation of the process proffered in the theory of natural selection, which was soon seen to throw a new and searching light upon the smallest details of structure, function, and distribution,—in fact, to afford the instrument of a new and systematic interpretation of organic nature. Of this doctrine, then, a brief account is needed, and this may be best given by following as closely as possible upon the lines of the magnum opus itself, although the full title of this— The Origin of Species by means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life— is its own best and briefest summary.

Theory of Natural Selection.

After mentioning that his first light upon the origin of species was derived from his early distributional studies, Darwin points out that “a naturalist, reflecting on the mutual affinities of organic beings, on their embryological relations, their geographical distribution, geological succession, and such other facts, might come to the conclusion that species had not been independently created, but had descended like varieties from other species. Nevertheless, such a conclusion, even if well founded, would be unsatisfactory until it could be shown how the innumerable species inhabiting this world have been modified so as to acquire that perfection of structure and co-adaptation which justly excites our admiration.” Again, “It is therefore of the highest importance to gain a clear insight into the means of modification and co-adaptation. At the commencement of my observations it seemed to me probable that a careful study of domesticated animals and cultivated plants would offer the best chance of making out this obscure problem. Nor have I been disappointed: in this and all other perplexing cases I have invariably found that our knowledge, imperfect though it be, of variation under domestication affords the best and safest clue.” Hence the first chapter is devoted to an account of those extensive studies of variation under domestication which later formed the subject of a separate work.

A comparison between individuals of the same variety of cultivated plants or animals shows a greater degree of variation than between individuals of any one species or variety in nature. This higher variability of domestic productions is to be ascribed to the less uniform conditions of their upbringing, perhaps in part also to excess of food. Exposure to new conditions must be continued for generations to set up any great variation; but this, once set up, continues indefinitely. Changed conditions may directly influence the whole organization or certain parts alone, or act indirectly through the reproductive system. With respect to the direct action, the nature of the organization seems more important than that of the conditions. The effect on offspring may be definite: e.g., size may depend upon the amount of food, colour upon quality of food, thickness of skin and hair upon climate, &c. But indefinite variability is a much commoner result of changed conditions, and has probably played a much more important part in the formation of our domestic races. The reproductive system is peculiarly sensitive to very slight external changes. Many plants and animals will not reproduce in domestication, even though individually vigorous; others, though weak and sickly, breed freely. Hence we need not be surprised at the reproductive system acting irregularly and producing variations. But that variation is not exclusively associated with sexual reproduction is demonstrated by the case of plants “sporting” through bud-variations. Such cases, moreover, prove the nature of the organism to be of more importance than the conditions.

Changed habits and use and disuse produce an inherited effect; witness the lighter wing-and heavier leg-bones of the domestic duck, the enlarged udders of milch-cows, or the drooping (unpricked) ears of domestic mammals. Variations are often definitely correlated: short-beaked pigeons have small feet; hairless dogs have imperfect teeth; and blue-eyed white tom-cats are deaf. Hence selection of any one character will probably modify others indirectly. Although the laws governing inheritance are mostly unknown, probably most, if not all, characters tend to be inherited. The popular statement that domestic varieties, when they run wild, revert to the primitive stock is unsupported by satisfactory evidence; and, although reversions occasionally occur in domestication, there is no sign of any general tendency to the loss of acquired characters. Except in being less uniform than natural species, in often differing more widely in some single part, and in being fertile when crossed, there are no well-marked distinctions between our domestic races and the so-called true species of a genus. The many breeds of dogs and cattle may arise from more than one species; but probably those of horses and fowls, and clearly those of rabbits, ducks, and pigeons, are each descended from a single species. In this respect the breeds of pigeons are of peculiar importance, since not only carrier and tumbler, runt and barb, pouter and fantail, but at least a score of varieties might be chosen which differ so thoroughly, internally as well as externally, that an ornithologist, treating them as wild birds, would be compelled to grant them specific, and even distinct generic rank. Yet, since all these have indisputably arisen from the wild rock-dove (Columba liυia), it is clear that naturalists who admit a unity to such domestic races, which professed breeders have often laughed to scorn, should in turn be cautious before deriding the unity of wild ones.

How then have domestic races been produced? By external conditions or habits alone? One of their most remarkable features is in exhibiting adaptation, not to their own good, but to man’s use or fancy. We know that all the breeds were not produced as perfect as we now see them, and the key is man’s power of accumulative selection: nature gives successive variations; man adds them up, making for himself useful breeds. Skilful breeders speak of the organization as plastic and under control, and have effected extensive modifications within our own generation. Unconscious selection, which results from every one trying to possess and breed the best individuals, is even more important. This accumulated change explains why we so often cannot recognize the wild parent stocks of our cultivated plants, while its absence in countries inhabited by uncivilized man explains why these never yield plants worth immediate culture. Man’s power of selection is facilitated by keeping large numbers, in which variations are more likely to occur. Facility in preventing crosses is also of importance, as of pigeons as contrasted with cats; some species are, however, less variable than others, e.g., the goose.

Individual differences arise even in the offspring of the same parents and tend to be inherited; hence they afford material for natural selection to act on and accumulate, precisely as they would for human selection. (Polymorphic genera— e.g., Rosa, Rubus, Hieracium, &c.—may perhaps owe their protean character to their variations being of no service or disservice, and consequently not being acted on by natural selection. ) In determining whether a form should be ranked as species or variety, the opinion of naturalists of sound judgment and wide experience is the only guide, yet this lacks unanimity (see Species) : for example, of the polymorphic genera of the British flora alone Bentham reckons 112 species but Babington 251. Wallace has shown that no certain criterion can possibly be given by which to define his convenient working categories of Malayan Lepidoptera, &c., viz., variable forms, local forms, sub-species, and representative species. As De Candolle concluded from his monograph on oaks (in which he shows at least two-thirds of his 300 species to be provisional), “so long as a genus is imperfectly known and its species founded upon a few specimens” they seem clearly limited; but, “just as we come to know them better, intermediate forms flow in and doubts as to specific limits augment.” The terms variety and species are thus arbitrarily applied to indefinable groups of more or less closely similar individuals. Common species that range wide and are much diffused are those which vary most. The species of the larger genera in each country vary more frequently than the species of the smaller genera. The species of large genera present strong analogies with varieties, which we can only understand if they originated as such.

The term “struggle for existence” is used in a wide sense, including dependence of one being upon another, and embracing (which is more important), not only the life of the individual, but success in leaving progeny. From the high (geometrical) rate of increase of all organic beings (slow breeders only requiring a few more years to people a whole district) struggle inevitably follows, either one individual with another of the same species, or with the individuals of a distinct species, or with the physical conditions of life. It is the doctrine of Malthus applied with manifold force to the entire animal and vegetable kingdoms, for in this case there can be no artificial increase of food and no prudential restraints from marriage. The checks to increase are most obscure. Eggs or young animals generally suffer most, and plants, mostly as seedlings, both from germinating on ground already occupied and from animals. The amount of food, of course, gives the extreme limit of numbers, very frequently, however, the attacks of enemies, as of game by vermin. Climate plays an important part, and periodical seasons of extreme cold have destroyed as many as four-fifths of the birds of an observed area. Epidemics, too, may occur. In many a species a large stock of individuals is often essential to its preservation. Complex and unexpected checks and relations exist between organic beings which have to struggle together; witness the profound alteration of the flora and fauna of a heath when planted with Scotch fir, these again being wholly dependent upon the exclusion of cattle. But in several parts of the world insects determine the existence of cattle. Again, red clover depends for fertilization upon the humble-bees, these upon immunity from the attacks of field-mice, and thus indirectly upon the numbers of cats; hence no bees, no clover, and the more cats, the more clover! The struggle will almost invariably be most severe between the individuals of the same species, for they frequent the same districts, require the same food, and are exposed to the same dangers. In the case of varieties of the same species the struggle will generally be almost equally severe, and we sometimes see the contest soon decided (as in the case of varieties of wheat or sweet pea, of the mountain-sheep or medicinal leech). Similarly,...

2. Again aψ(x) is the difference of the y ordinates of the two curves for the same value of x; this indefinitely small difference is called the variation of y, and is denoted by δy. It may be regarded as the change in y arising solely from a change in the relation connecting y with x, while x itself remains unaltered. In general, if y be a function of any number of independent variables x 1 , x 2 ,. .. x n , then δy represents any indefinitely small change in y arising solely from a change in the form of the function, while x 1 , x 2 , &c., are unchanged. Thus the variable y may receive two essentially distinct kinds of increment,—one arising from a change in one or more of the variables, the other arising solely from a change in the relation which connects y with these variables. The former increments are those contemplated and treated of in the ordinary calculus; the latter are those principally considered in the calculus of variations.

We shall follow Strauch, Jellett, Moigno, and the principal modern writers on the subject, by restricting, in general, the symbol δ to the latter species of increment. In many problems both kinds of increment have place simultaneously. Thus, if y =f(x 1 , x 2 , x 3 ), and if ∆y denote the total increment of y, we have = + (2); and so on in all cases. g y g

3. It is readily seen that = and in general jn y ç» β that δ y j ζ = ¾Λ The last equation may be written δD n y=D n δy . (3), where D stands for the symbol of differentiation, d/dx .

4. We shall adopt Newton’s notation and write y for dy/dx, ÿ for d 2 y/dx 2 , y(n) for d n y/dx n and proceed to consider the variation of the general expression V =f(x, y,y,ÿ,... y n ), in which the form of the function f is given, while that of y in terms of x is indeterminate. Here, considering x as unchanged, we have

Now let then we have δ V~= Pδy + P v Dδy + P⅛D~δy + . . . +P n D n δy (4).

5. Next let us consider the variation of the definite i xι integral Z7 = I Vdx, where V does not contain either of the limits x 0 and x 1 . Here, when the limits are unchanged, we have 8U= I δ Vdx, and, when the limits undergo change, J *0

δU=δ∣ 1 Kdx= ∣δVdx+ V 1 dx 1 - V 0 dx 0 (5).

J xo J xo

6. We shall here bring in a new symbol, called the sign of substitution, which was introduced by Sarrus. Thus, if F be any function of x, y, &c., the result of substituting x 1 for x in F —the other variables being unchanged—is

∕ xl F in the form of notation adopted by ∕ x l

Moigno and Lindelöf. The difference between ∕ F and ∣ x ^F is represented by J 1 F. Employing this notation, (5) is written δU= ∣ δVdχ+ ∕ Vdx (6).

J xo ∕ XO

7. We now proceed to transform the last equation, commencing with the case in which V is a function of x, y, and y solely.

∕⅛1 ∕≈ι

Here δU= I <J>δy + P v Dδy}dx+I Vdx, -h F - d ·^ + /y 9+ /^ m

Next let V be a function of x, y, ÿ, and ÿ, then

δ U= Γ (Pδy + Pι∑>δy + P 2 B 2 δy)dx + ∕ Vdx.

Hence, observing that we get W=∫^(i>-⅛ i +⅛)⅛'fe+∕ j .' r *'

+ ∕^ λ -S) s ≈' + ∕> λ (8) ·

In general, if V be a function of x, y, y, ÿ,. . . y n we have δU= Γ∖pδy + P 1 Pδy + . . . + P n D n δy)dx + ∕ 1 Vdx. J a⅛ ' a ⅛ . .

Now let D 1 denote the differentiation of δy, and D 2 the differentiation of P 1 , P 2 , &c.; we may write P m D m δy - ( - 1) m δyD m P m =(D 1 m - ( - 1) m D 2 m )P m δy = (D 1 + D 2 )(D 1 m-1 -D 1 m-2 D 2 + .... ±D 2 m-1 )P m δy

Hence [ 11 p m D m δydx={- l) m f 1 ^^δydx J x 0 J x 0 dx + <- ι )"-⅛^⅛)∙ Applying this to the different terms in the value of δ U given above, " egβt

√>-⅛÷∙∙∙÷<-υ"-≤⅜> ÷∕>S÷∙∙∙÷<-^)⅛ √>5⅛t∕> w ∙

This result may be written

ΛT 1 ,x 1 δU= ∕ {P)δydx+ ∕ Vdx J X 0 'Xo +∣ x ∖{P 1 )δy+{P 2 ')δy+ ... +(P n )δ√ n - 1 >} (10).

when, (P)=P-⅛√⅛- .... (P,)= λ -¾ + ⅛- ..„and so on.

8. This expression consists of three parts,—(1) the de-

P 1 finite integral I (P)δydx, which depends on δy, the change J XO Vdx, x 0 which depends solely on the change in the limiting values of x; and (3) the quantity ∕ 1 {(Z∖)δy + (P 2 )δ^ + · · · } , ' ⅜

which depends on the variations of y, y, &c., at the limits. It is often convenient to write (10) in the abbreviated form δU=L + f Mδydx (ll).

J xo

9. The principal applications of the calculus of variations have reference to the determination of the form of one or more unknown functions contained in a definite integral, in such a manner that the integral shall have a maximum or minimum value. For instance, to determine the form of the function y which renders f Vd jc a maximum or a minimum, we have as above δU-L + ^ 1 Jfδydx. Here, as in the differential calculus (see Infinitesimal Calculus, § 64), for a maximum or minimum value of U it is readily seen that we must have δ U= 0; this leads to Γx 1 L = 0 and I Jfδydx = 0. Now the latter integral cannot J xo . be zero for all indefinitely small values of δy unless M = 0 for all values within the limits of integration. Hence we get the differential equation in y,

<>’>·

10. We here suppose that there is no restriction on δy, so that for any value of x the increments + δy and - δy are equally compatible with the conditions of the problem. The reasoning consequently will not apply if the conditions render this impossible. For instance, if a curve be restricted to lie within a given boundary, then for all points on the boundary the displacements must be inwards, and the opposite displacements are impossible. In this case the curve satisfying a required maximum or minimum condition consists partly of portions of the boundary and partly of portions of a curve satisfying the equation M= 0.

11. The equation M =0 is in general a differential equation of the degree 2n; accordingly its solution usually contains 2n arbitrary constants. The values of these constants are to be determined by aid of the equations deduced from L =0 in combination with the given conditions at the limits. For example, if V be a function of x, y, dP and ÿ solely, the solution of P--^=0 is in general of the form y=f(,x, c 1 , c 2 ). Now, suppose that the limiting points are restricted to the curves y 0 =f o (x o ) and y 1 = f 1 (x 1 ), then we get at the limits δy0+y0dx0=f 0 '(x 0 )dx 0 and δy 1 + y l dx l =f 1 '(x 1 )dx 1 . Hence, substituting in L = 0 the values of δy 0 and δy 1 derived from these equations, we get, since dx 0 and dx 1 are arbitrary,

F"o + (∙Pι)oC∕o ’ (⅜) - ⅛ ’ o) = θ∖ ∕'i¾'(

P 1 +(Λ)ι(ΛX¾)→ι)=oΓ∙∙÷ ( λ

We have therefore six equations for the determination of the six quantities x 0 , y 0 , x 1 , y 1 , c 1 , and c 2 .

The integration of the equation M =0 is much simplified in particular cases. -

(1) Let V be a function of x and y solely, then we have =0, therefore P 1 = c (14).

(2) If V be a function of y and ÿ solely, P=⅛⅜, also ⅛ V) = Pÿ + P 1 jf dP dx dx

= -τ⅛' + P 1 y. Hence, integrating, we get V=P 1 y + c (15), where c is an arbitrary constant.

12. For example, let us consider Bernoulli’s problem of the curve of quickest descent under the action of gravity. Take the axis of x vertical and that of y horizontal, and suppose the particle to start from the point x 0 , y 0 , with the velocity due to the height h; then, if v be the velocity at any point, we have v 2 =2g(x + h-x 0 ), also dt= —= ..... Here, neglecting a constant factor, we may v √2<7√z + ⅛-ar 0 & J write Z7=; and by ( 14 ) we get - =c V¾ + hx„. jχ o ^dx + h-x o , ∖L+y

Now, writing a for l∕c 2 and assuming x + h - x 0 = a sin 2θ , we find dy =tan θdx=a(1 - cos2θ)dθ ; therefore y=a(θ - sin θ cos θ) + K, and we infer that the curve is a cycloid.

(a ) If we suppose the upper limit to be fixed and the lower restricted to the curve y 1 = f 1 (x 1 ), we get from (13)

l + tanθ 1 f 1 '(x 1 )=0 (16); accordingly the curve intersects the bounding curve orthogonally. (β ) Next let the upper limit be restricted to the curve y 0 = f 0 (x 0 ); then since V contains x 0 , we have an additional term in δU arising from C x ^d V the change in x 0 , viz., dx 0 I dx ; consequently the coefficient of J x o clx o dx 0 in δU is f'⅛⅛x - {F o + (PM ’ { χ o) ~ ^o)}, f x ιdV and we have J ^d χ = f r o + (Λ)o(∕o'(a ⅛) ~ ⅛ ’ o) (17)∙

If we substitute for V its value, we get without difficulty from this equation 1 + tan θ 1 f 0 '(x 0 ) = 0. Comparing this with (16), we see that the tangents to the bounding curves at the extremities of the trajectory are parallel (Moigno, op. cit., p. 230).

13. More generally, consider the curve for which U— ί μds rχι . J xo = Ι ΛV1 + i> 2 dx is a maximum or a minimum, where μ rs a given

* ∙^0 d∣JL U.Ï/ function of x and y. Here P= ~r∖∕l + z∕ 2 , P- l =— ==λ- , and we get ⅜ 1 √Γ+Z 6 dμ Λ ίdμ .dμ∖ d ( ÿ ∖ + ÿ2= √i7FΛ^ ⅞∕ ; hence' if ρ be the radius of curvature, we have

Γ-J(cos ⅛ +c,,s ^t) < 18 >’

where α, β are the angles which the normal makes with the coordinate axes respectively. Jellett remarks that, if the proposed /ds ·

• , we should then have arrived at the equation

- = -i cos a, c ^ + cosS⅛Y and consequently have seen that the two P μ∖ dx dyj curves contained under the equation ρ 2 =f(x,y,y) are such that, if one renders J^ μds a maximum or a minimum, the other possesses the same property with regard to (—.

14. Next, to find the curve such that the surface generated by its revolution round a given line shall be a minimum. Here, neglecting a constant multiplier, we may write U = I yVl + y i dx;

J XQ

accordingly by (15) we have y =c√1+ y 2 . This gives -, —- = . ∖∕y--c i c , , x+b 1 ∕ ∖⅞ 2 -c 2 ∖

and hence —= log (yd —— 1, or y = c -^+b∙)∣c +e -(x+b)l^ (19), which represents the common catenary.

15. We now proceed to generalize the results in § 11. If V be a function of x, ÿ, ÿ, . .. y {n not containing y explicitly, we have by integration

Λ-S÷⅛-∙÷<-υ-^ -

where c is an arbitrary constant. Again, if V does not contain x explicitly, ^( Γ)= ÿP+yP 1 + ... + ^ ft+1 ^P n ∙ Also, from the equation 3f= 0 we see that P=^- 1 -⅛⅛ + ... - ( -1)" ςτ~p; hence dx dx i dx n =(D 1 + Z) s X⅛P 1 ) + (2√ - Λ 2 )(^<s) +... + (Λ" - ( - 1 WW,), where D 1 represents the differentiation of y, ÿ, &c., and Z> 2 that of ∕∖, P 2 , &c. Consequently, by integration

Γ=c + 2 ∕(P 1 ) + ÿ(P 2 ) + . ·. + √ 7l >Λ 1 (21), a differential equation of the order 2n - 2 at the highest. Again, if V contains neither x nor y explicitly, we may substitute c' for (P 1 ), and consequently we have

V=c + c'⅛+y(P 2 )+ ... +√ n >P n (22).

16. As an example, let us investigate the curve for which the area between the curve, its evolute, and the extreme radii of curva-fs 1 ∕∙¾(1 + λ2)2 ture shall be a minimum. Here U= pds= — -d x. Hence J √ J χ 0 y by (22), since yP 2 = - V, we have P^=c 1 + c i f∣, or p = . C1 - -1—-=; ds . Vl + ÿ 2 VI + y 1 therefore 5 - = c 1 cos <∕> + c 2 sin ≠, where φ is the angle made by the dφ tangent with the axis of x. This leads to s= -c 1 sin ≠ + c 2 cos φ + c 3 (23), which shows that the curve is a cycloid. Again, (1) if the extreme points be fixed, the equations at the limits are (P 2 ) 0 = 0, (P 2 ) 1 = 0. This shows that the radius of curvature vanishes at both limits and the curve must be a complete cycloid. (2) If the extremities lie on the curves y 0 -∕o(⅞)> Vι=fι( x ι) ! then the equations furnished by L =0 are Po + (P])ol∕o'(⅜) - ⅞i = θ> (Ρ2)ο = θ1 ι<)i∖ i r ι + (P 1 ) 1 W(a∙ 1 )~⅛ι} =0, (P 2 ) 1 = 0 f

Hence the extreme points are cusps on the cycloid; moreover, since in this case V o = 0 and V 1 = 0, we have y 0 -∕ 0 '(a⅛) = θ> ⅛ 1 -∕ 1 '(αq) = 0, which show that the cycloid at each of its extremities touches these curves, and that the line joining the extreme points is normal to each of the bounding curves.

It is easily seen that the minimum area is four times that of the circle which generates the cycloid. vι

17. Next let us consider the variation of £7= I Vdx, J Xo where V is a given function of x, y, 2, ÿ, 2,y,2,... y^ n ∖ zt m ∖ and y, z are undetermined functions of x. τ . λ dV n dV r ,

Ut ' Q= li ,Q t =- 3i ,l u1 ., then SU= ∕ 1 ^Pdx+ ί (Pδy + P 1 Dδy +.. ,+P n IPSy)dx

'Xo J x o

Γ x ι

+ I (Qδz+Q 1 Dδ2 + . . . +Q m D δz)dx. J zθ

Proceeding as in § 7, we find δU= Γ∖p)δydx + Γ 1 (Q)δzdx + ∕ xi Vdx

J Xn J Xλ I x a

+ ∕ xι {(P 1 )δy+(ΛW+· · · + 7 ’ n δ√ n - υ }

∕ x 0 + / X1 {(Qι)Sz+(Q a )δz + ...+ Q m δz<∙ m - 1 >} (25), ' Xo where (7 1 ), (Z ’ 1 ),... have the same meaning as in § 7, and (Q ), (Q 1 ),.. . are the corresponding functions for the variable z . r χ ι

18. The determination of y and z where 1 Vdx is a χ o maximum or a minimum leads, as in § 8, to the equations (P ) = 0, (Q)=0, along with the equation at the limits ∕Vd⅛+ ∕* 1 {(P 1 ')δy + ...P n δ∕ n ~ γ >} + ∕ x ∖(Q 1 )δz+ ...Q m δJ m ~V} =0. I Xo ' Xo ,χ o

The mode of treatment is similar to that for a single dependent variable.

19. In the discussion of a curve which possesses a maximum or minimum property, if we limit the investigation to all curves of a given length or which satisfy some other condition, we have a distinct class of problems, which originated in the isoperimetrical problems of James Bernoulli. They were originally styled isoperimetrical, but are now called problems of relative maxima and minima. Thus, let it be proposed to determine the form of y which renders c χ ι U = I Vdx a maximum or a minimum, and which also J Xo f x l satisfies the condition V = I Vdx = constant, where V and j Xo V are given functions of x, y, ÿ, &c. It is obvious that, if U is a maximum or a minimum, so also is U+ aU', where a is any arbitrary constant. Accordingly the problem reduces to the determination of the maximum or minimum value of the integral f (V + aV')dx, regarding a as a constant, whose value is to be determined by aid of the given value of U'.

20. Another class of problems closely allied to the preceding is that in which the variables are connected by one or more equations of condition: for instance, when y, ÿ, ÿ, & c., z, z , &c., are connected by a relation W= 0, we investigate the maximum or minimum values of U=[ xι (Γ+λJV)dx (26), J xo where λ is an indeterminate function of x. We will illustrate these principles by examples.

21. To find the curve connecting two fixed points such that the surface generated by its revolution round a fixed line shall be given, and the volume of the generated solid shall be a maximum. Taking the fixed line as the axis of x, we may write U= (y 2 + ay∖ll + y∙ i )dx. J X o

Hence by (15) we have 1∕ 2 + αy√l+⅛ 2 =c + ^p==,

ΛV

or ^ + √γ⅞f= c -' ' 2i, ∙

If the curve meets the fixed axis we have c =0, and the curve is a circle whose centre lies on the axis.

The further integration of (27) depends on elliptic functions; it can, however, be shown without difficulty, as was proved by De-launey, that the curve is that generated by the focus of...