Gravitation

GRAVITATION

I. Gravitation at the Surface of the Earth.

IT is a matter of universal experience all over the earth that a heavy body tends to fall to the ground. Let us inquire into this in the first place by taking such a general view of the phenomenon as would be presented to an imaginary spectator who was sufficiently removed from the earth to be able to take a general view.

Fig. 1 represents a section of the earth by a plane which is drawn through its centre 0. Then, the earth being sufficiently near a sphere for our purpose, we may regard the section PQRS as a circle, where the points P, Q, R, S are the intersections of the lines 0A, OB, 0C, OD with the surface of the earth. If a stone be dropped from a point A above the surface of the earth, it will fall to the ground at P. The spectator would also notice that, if a stone were dropped from B or C or D, it would fall upon the ground at the points Q, R, S respectively. From A the stone would appear to fall downwards, from C it appears to move upwards, from B the spectator would see the stone moving to the left, while from D it appears to move to the right. One feature of these motions could not fail to be noticed: they all tend to the centre of the earth. The spectator might therefore sum up his experience in the following statement :—

J body dropped from a point above the surface of the earth always falls in a straight line which is directed towards the centre of the earth.

§ 1. Attraction.— The familiar instance of the action of a magnet upon a piece of iron will suffice to illustrate what is meant by the word attraction. In virtue of certain properties possessed by the iron and the magnet, they are drawn together. The magnet draws the iron, and the iron draws the maguet This particular kind of attraction is of a very special character. Thus, for example, the magnet appears to have no appreciable influence on a piece of wood or a sheet of paper, and has indeed no considerable influence on any known substance except iron.

By the attraction of gravitation, every body attracts every other body, whatever be the materials of which each is composed. In this we see a wide difference between the attractions of gravitation and that form of attraction which is known as magnetic attraction. Nor is the contrast between the intensities of these two different attractions less striking. The keeper of a magnet is drawn to the magnet by two different forces of attraction. The first of these is the gravitative attraction, which, so far as we know at present, would be equally exerted, whether the magnetism were present or not The second is the magnetic attraction. The latter is enormously greater than the former; in fact, under ordinary circumstances as to intensityand dimensions, the intensity of the attraction of gravitation will not be nearly so much as a millionth part of the magnetic attraction.

The intensity of the attraction of gravitation is indeed so small that, with one conspicuous exception, we can only become aware of its existence by refined and elaborate inquiries. That any two objects—for example, two books lying on the table—do actually attract each other, there can be no doubt whatever; but the intensity of this is so small that the attractive force cannot overcome the friction of the table, and consequently we do not find that the books are drawn together. It has, however, been found that the intensity of the attraction of gravitation between two masses is directly proportional to the product of those masses. Hence though the force is so small as to be almost inappreciable between two bodies of moderate dimensions, yet when the masses of the two bodies, or of even one of them, are enormously great, the intensity of the force will be sufficiently large to be readily discernible. In this way it is that the existence of the attraction of gravitation has been made known to us, and is, in fact, identified with our daily experience,—indeed, with our actual existence. The mass of the earth is so enormous that the attraction of gravitation which exists between it and an object near the surface is readily appreciable. It is this attraction of gravitation between the earth and any object which constitutes that force which is referred to when we speak of the weight of the body. It is the attraction of gravitation which causes bodies to fall to the surface of the earth; and it is easy to show that the facts already presented with respect to the direction in which a falling body moves are readily explained by the supposition that the motions are due to an attractive influence exerted by the earth, or, to speak more correctly, to a mutual attraction subsisting between the earth and the body.

Let 0 be the centre of the earth, supposed to be a sphere, and let A be the position of a body above its surface. Then, when the body is released, the attraction must evidently cause the body to move along the line OA. The line OA, in fact, is directed along a diameter of the sphere, and there is really no reason why the stone should move te one side of the line more than to another. We thus see that attraction would always tend to draw objects in a direction poiuting towards the centre of the. earth. The observed facts are therefore explained by the supposition that the earth possesses a power of attraction.

§ 2. Movement of a Falling Body.— Our knowledge of the force of gravitation being ultimately founded on observation and experiment, it will be convenient at this point to describe the experiments by which a knowledge of the laws of motion of a falling body may be ascertained. We shall first describe these experiments, and then we shall discuss the laws to which we are conducted by their aid.

A beginner is apt to be surprised when he is told that a heavy body and a light body will fall to the ground in the same time if let drop from the same height.. Yet nothing can be easier than to prove this important fact experimentally. Take a piece of cork in one hand and a bullet in the other, and drop these two objects at the same moment from the same height. They will reach the ground together. Nor will the results be different if we try a stone and a piece of wood. If, however, one of the objects were a feather and the other were a stone, then no doubt the latter would reach the ground long before the former. But this arises from a cause quite different from gravity. It is the resistance of the air which retards the motion ol the feather. Even the stone is retarded to a certain extent by the resistance of the air; but the feather, on account of its greater surface in proportion to its mass, is much more retarded. If we could get rid of the influence of the air, the stone and the feather would be found to fall to the ground in the same time. This can actually be verified by performing the experiment (or a similar one) in a space from which the greater portion of the air has been withdrawn by the aid of an air-pump. But the same thing can also be shown in a much more simple manner.

Lay a small flat feather upon the top of a penny piece held horizontally. Then let the penny fall; it will be followed with equal rapidity by the feather, which will be found to remain in contact with the penny throughout the entire descent. In this case the penny piece displaces the air, and thus to a great extent shields the feather from the resistance to which it would be exposed without such protection; it is thus found that the two objects fall to the ground from the same height at the same time.

The various experiments to which we have referred suffice to establish the very important result that the time occupied by a body in falling to the surface of the earth, if dropped from a point above it, is independent of the mass of the body as well as of the materials of which the body is composed.

There are, no doubt, certain apparent exceptions to the generality of this statement. The law, as we have stated it, does surely not apply to the case of a balloon or a live bird. In each of these cases the air is made, directly or indirectly, to supply a force which overcomes the force of gravity and neutralizes its effects; but if there were no air, then the balloon and the bird would fall to the ground in precisely the same time as a 56 δ) weight would do when dropped from the same height. It will not be necessary for us to introduce any further reference to the resistance of the air, and we shall discuss the phenomena presented by falling bodies as they would occur in a space from which the air has been removed.

We have by these considerations cleared the way for a very important quantitative determination. Taking a given interval of time,—for example, one seoond,—we see that the height through which a heavy body will fall in one second depends neither upon the mass of the body nor on the materials of which it is composed. This is therefore a constant at any given place on the earth’s surface for every description of body, and it is of fundamental importance to determine that quantity accurately. By an indirect method, founded on pendulum observations, it is possible to determine this quantity with far greater accuracy than would be attainable by actually making the experiment. The value as thus found is slightly different at different parts of the earth though constant at each one. At any part of the United Kingdom it may be taken as 164 feet.

When the distance which the falling body moves over in the first second has been ascertained, it is possible to find the distance which will be accomplished in two seconds, or indeed in any number. The difficulty of the question arises from the circumstance that, as the velocity of the falling body is gradually increasing, the distance moved over in the second second is greater than it was in the first, and generally that the distance in any second is greater than the distance accomplished in any previous second. Imagine the “lift” in a hotel to be a room 164 feet high; then when the lift is at rest, a stone will take one second to fall from the top of the room to the floor. But now suppose the experiment to be repeated, when the lift is either ascending or descending. It will be found that no matter what be the velocity of the lift, provided it remains uniform for a second, and no matter whether the lift be ascending or descending, the stone will still take exactly one second to fall from the ceiling to the floor.

To illustrate the important conclusions which can be drawn from this experiment, let us make some suppositions with reference to the velocity of the lift. Suppose that the lift is descending with a velocity of 5 feet per second. Then since it is found that the stone will reach the floor in one second, it is manifest that during that second the stone must actually have fallen through a distance equal to the height of the room augmented by the 5 feet through which the floor of the room has descended. The total distance traversed by the stone is therefore 16 ’ l + 5 = 21∙1 feet. It is, however, to be observed that at starting the stone must necessarily have had the same velocity as the lift, i.e., 5 feet per second. The observed facts can therefore be explained by supposing that the stone retained its initial velocity of 5 feet per second, and that gravity acted upon the stone so as to draw it 16∙1 feet nearer the earth than it would have been had gravity not acted. On the other hand, suppose that at the time when the experiment was made the lift was ascending with an uniform velocity of 5 feet per second. Then the actual distance travelled by the stone in falling will be less than the height of the ceiling by the distance through which the floor has been raised, i,e., 16 ’ 1 - 5 = 114 feet. Observation nevertheless shows that the time occupied in falling from the ceiling to the floor is still one second. The observed facts can be explained by remembering that at the moment of starting, the stone must actually have had the same velocity as the lift, i.e., an upward velocity of 5 feet per second. If therefore gravity had not acted, the stone would in one second have ascended through a vertical distance of 5 feet. The observations are therefore explained by supposing that gravity in this case also draws the body 164 feet nearer the earth in one second than the body would have been had gravity not acted.

By suitable contrivances it is possible to ascertain that a body dropped from rest will in a time of two seconds move over a space of 64 , 4 feet. We have already seen that during the first second the body will fall 164 feet. It follows that in the second second the space described by a body falling freely from rest is 64 - 4 - 164 = 48 ’ 3. It is thus obvious that the space described in the second second is three times as great as the space described in the first second. To what is this difference to be ascribed 1 At the commencement of the first second the body was at rest; at the conclusion of the first second the body had attained a certain velocity, and with this velocity the body commenced its motion during the second second. The total distance of 48 ’ 3 feet accomplished during the second second is partly due to the velocity possessed by the body at the commencement, and partly to the action of gravity during that second. By the principle just explained, we are able to discriminate the amounts due to each cause. It appears, from the experiments already referred to, that during the second second as during the first the effect of gravity is simply to make the body 164 feet nearer the earth than it could otherwise have been. But the body moves altogether 48 ∙3 feet in the second second, and as the action of gravity during that second will only account for 164 feet, it follows that the residue, amounting to 48 ’ 3 - 164 = 32 ’ 2 feet, must be attributed to the velocity accumulated during the first second.

We are therefore led to the very important result that a body falling freely from rest in the United Kingdom will have acquired a velocity of 32 2 feet per second when one second has elapsed. It need not be a matter for surprise that, though at the close of the first second the velocity acquired is 32'2, the distance moved over during that second is only 164. It will be remembered that the body starts from rest, and that while in the act of falling its velocity is gradually increasing. The body, therefore, moves much further in the last half of the second than it did in the first half, and consequently the total distance travelled must be lese than the distance which would have been accomplished had the body been moving during the whole second with the velocity acquired at its termination.

It might not be easy to arrange a direct experiment to show how far the body will fall during the third second; we can, however, deduce the result by reasoning from what we have already learned, Let us suppose that the lift already referred to is descending with an uniform velocity of 32'2 feet per second. A body let drop from the ceiling during the motion, will, as before, reach the floor in one second. The body will therefore have acquired, relatively to the moving lift, a velocity of 32 , 2 feet per second. But the lift is itself in motion with a velocity of 32'2 feet per second. The actual velocity of the body must be measured by its velocity relatively to the lift·, added to the velocity of the lift itself. It therefore appears that the body which, when it commenced to fall, had a velocity of 32 , 2 feet per second, acquires an equal amount during its fall, so that at its close the body actually had a velocity of 32 , 2 + 32 - 2 = 64 , 4 feet per second. A body falling freely from rest acquires a velocity of 32 ’ 2 feet in the first second; it follows that at the close of the first second the body is in the same condition as if it were let fall from the ceiling of the lift, under the circumstances just described. The motion during the third second is therefore commenced with the velocity of 64 ’ 4 feet, and in consequence of this initial velocity alone a distance of 64 - 4 feet will be accomplished in the third second. To this must be added 16T feet, being the additional distance due to the action of gravity, and therefore we have for the distance through which a body falling freely from rest will move in the third second, 64 ’ 4 + 16T = 80 ’ 5. Similar reasoning will show that the velocity acquired at the close of the third second is 64 ’ 4 + 32 ’ 2 = 96 ’ 6. With this velocity the fourth second is commenced, and therefore the distance accomplished during the fourth second is 96'6+ 16 1 = 112 7.

The results at which we have arrived may be summarily stated in the following propositions :—

A body falling freely from rest acquires a velocity which is equal to the product of 32 ’ 2 and the number of seconds during which the motion has lasted. A body falling freely from rest moves over spaces proportional to the consecutive odd numbers (1, 3, 5, 7, &c.) in each of the consecutive seconds during which the motion lasts. A body falling freely from rest will, in a given number of seconds, move over a distance which is found by multiplying the square of the number of seconds by 16 ’ 1.

§ 3. Values of g.— The velocity acquired by a body in one second is usually denoted by the symbol g. The following are values of g at different parts of the earth (adapted from Everett On C.G.S. Units, p. 12):— [table]

The value of g in feet at a station of which the latitude is λ, and which is h feet above the level of the sea, is in feet g = 32 ∙173 - 0 ∙082 cos 2λ - 0 ∙000003Λ.

The length of the pendulum in feet which vibrates in one second is l = 3 ∙2597 - 0’0083 cos 2λ - 0 O00000⅛.

g is really the excess of gravitation over the centrifugal force arising from the earth’s rotation. The value of gravitation alone is given by the following expression :—

32∙225-0∙026 cos 2λ.

§ 4. Algebraical Formulae.— The employment of the symbols and operations of algebra will enable us to express very concisely the results at which we have arrived.

Let V denote the velocity acquired in t seconds by a body which has been dropped from a state of rest. Let s denote the number of feet over which the body has moved. The laws we have arrived at may be thus expressed :—

f-ÿi; s=Bgt 2 .

From these equations we can eliminate i and obtain v 2 = 2gs.

This expresses the velocity acquired in teπns of the distance through which the force has acted.

We have hitherto considered the movement of a falling body which was simply dropped. It remains to determine the effect on the movement of the body which would be produced by a certain initial velocity. Let us for simplicity take the case of a body thrown vertically downwards, and calculate the distance through which the body will move in a certain time, as well as the velocity which it will acquire. In the act of throwing the hand moves with a certain velocity, and the body when released starts off with that velocity. It will thus be observed that the act of throwing is merely to impart initial velocity to the body. Let v , be the initial velocity with which the body leaves the hand. Then the velocity of the body at the moment of starting is precisely the same as it would have been had it been dropped from rest v'÷g seconds previously. The velocity acquired at the end of t seconds is therefore the same as would have been acquired by a body which fell from rest for a period of if ÷g + t) seconds; whence we have v=gf÷g+t) = υ' + gt.

The distance must obviously be equal to the difference between the distance through which the body would drop from rest in (√÷g + i) seconds and the distance through which a body would drop from rest in v' ÷g seconds; whence

β- ’ 5giv' ÷g+t) 2 - ∙5g(d÷f) i - ∙5g(ι∕ 2 ÷g 2 + 2v ’ t÷g + t 2 ) - ∙5g(υ' 2 ÷g 2 ) =v't + '5gt 2 .

The case of a body projected vertically upwards seems at first to present somewhat greater difficulties, but this is not really the case. Such problems can always be readily solved by the help of the following general principle :—

A body moving vertically for t seconds will, at the end of that time, be '5gt 2 feet nearer the earth than it would have been had gravity not acted.

If the body be projected vertically upwards with an initial velocity v', then, if the influence of gravity were suspended, the body would in t seconds ascend to a height v't in accordance with the first law of motion. The effect of gravity will be to reduce the height actually obtained by the amount 'bgt 2 . Whence we have s=√i- ∙ζ>gt 2

This expression may be written in the form s=√ i ÷2<z - ∙δg(t - v'÷g) 2 .

It is therefore obvious that the greatest altitude h is attained when i = √÷gr; in which case h = v' 2 ÷2g, or v' i = 2gh.

As an illustration we may take the case of a body thrown vertically upwards with an initial velocity of 40 feet per second, and inquire where that body will be at the end of two seconds. Had gravity not acted, the body would, in two seconds, have ascended to a height 2 × 40 = 80 feet. The action of gravity will reduce this by ∙5<∕2 2 = 2g feet, and hence the actual height of the body will be 80 - 2g feet, = 15’6 if g be taken at 32‘2.

§ 5. Motion of a Projectile.— We have hitherto referred only to the motion of a falling body in a vertical line; it will now be necessary to examine some cases in which the motion of the body is not so restricted. From a point on the mast of a steamer g÷2 feet above the deck a ball is dropped, which falls upon the deck at a certain point. When the steamer is at rest the time taken by the ball to fall will of course be one second, and its path will be vertical; but when the steamer is moving with uniform velocity it is found that the ball still falls precisely on the same spot of the deck as when the steamer was at rest, and that the time occupied in the descent is still one second. It is obvious that in this case the ball does not move in a straight line at all, but in a curved path due to the motion of the vessel compounded with the actual falling motion. We therefore see that the effect of the motion of the vessel on the ball was to project that ball with a certain initial horizontal velocity, but that notwithstanding that initial velocity the ball still reaches the deck in one second. We are therefore led to the general conclusion that A body projected horizontally will, at the end of t seconds, have fallen through a space of 16 - 1 feet.

An experiment illustrating this result may be made in an exceedingly simple manner. Take a marble in each hand, and throw one of the marbles horizontally at the same time as you drop the other from the same height; you will find that the two marbles reach the ground together. Suppose for simplicity that the height at the moment when the marbles are released is 4 feet, then the time taken by one of the marbles in falling is half a second. But as both marbles reach the ground together, the experiment has really proved to us that a body at the height of 4 feet from the ground will if projected horizontally reach the ground in half a second. This is equally true whatever be the magnitude of the velocity, t.e., whether it be 5, 10, or any other number of feet per second.

We have now studied the effect of gravity upon a body which has been projected either in a vertical line or in a horizontal line. We have found that in each case the effect of gravity is to bring the body '5gC 1 feet nearer the surface of the earth in t seconds than it would have been had gravity not acted. We are therefore tempted to inquire whether the same statement would not be true for a body projected in any direction. In every way in which this suggestion can be tested it has been found to be verified, and there cannot therefore be the slightest doubt that it is true.

To illustrate this principle we may apply it to the case of a body projected in any direction, and deduce the form of the path in which the body moves. Let 0 (fig. 2) be the point from which the body is projected, and let OP be the direction in which the body would move after projection if it were not for gravitation. In consequence of the first law of motion, we should find that if it had not been for the action of gravity the ball would reach P 1 in one second, P 2 in two seconds, &c., where OP 1 =P 1 P a = P.P 3 ... Gravitation will, however, make the body swerve from the direction OP 1 P 2 , &c., so that at the end of one second the body is really found at the point A 1 , at the end of two seconds at A 2 , at the end of three seconds at A 3 , &c. The curve drawn through the points A 1 , A 2 , A 3 , &c., which is actually described by the body, can be readily constructed. Take, for example, i=3. If gravity had not been acting, the body would in three seconds have reached the point P 3 . We can find where the body actually is by taking a point A 3 , which is vertically beneath P 3 at the distance 16 T × 3 2 feet. Similarly we can find where the body is after any other specified number of seconds, and thus we obtain the points A 1 , A 2 , &c.

The equation of the curve is thus found. Take the line OP as the axis of x, and let x denote the number of seconds during which the motion has lasted; then, if y denote the vertical distance through which the body has been deflected by gravity, we must have y = v'x- '5gx 2 .

This curve, being of the second degree, represents a conic section; and as the highest terms form a perfect square, the conic section must be a parabola.

II. The Centre of Gravity.

§ 6. Centre of Gravity of a Plate.— In studying the effect of gravity upon a body which is at rest, it will be convenient to commence with a simple illustrative experiment which can be easily tried. Out of a piece of cardboard or tin plate a figure of any shape, ABODE (fig. 3), is to be cut. A few holes, A, B, C, D, E, are to be punched quite at random in this plate. In the wall is a nail, and the plate can be suspended by passing the nail through any of the holes A, B, and C. From the nail is suspended, in front of the plate, a cord AH, which is kept in the vertical direction by the plummet attached to it at H. As the plate is not supported in any other way, it hangs quite freely from the nail; and if it be displaced and then released, it will, after a few oscillations, settle down again in the position which it occupied at the first. In order to mark this position, it is desirable to draw a line on the plate in the direction AP, indicated by the plummet line which is hanging in front. If the plate be blackened, this can be neatly done by chalking the plummet line and then giving it a flip against the plate.

When the line has been drawn the plate maybe removed from the nail, and again suspended by one of the other holes in its margin, for example B. The plate again assumes a definite position, and again the line and plummet is to be hung on, and a second line drawn as before. The two lines drawn on the plate intersect at a point P. When the plate is hung from a third hole, C, and a third plummet line is drawn, a very remarkable result is perceived. It is found that the line drawn on the third occasion passes through the intersection of the two former lines,— that is to say, all the three lines pass through the point P. Repeating the operation, with other holes, D, E, &c., it is found that all the lines drawn in the way we have described possess the remarkable property of passing through one definite point of the plate. It is therefore manifest that the point P possesses a very special property, for it is always situated vertically beneath the point of suspension when the plate is hanging at rest.

If a hole be actually punched at the position of the point P, and if the plate be suspended by passing this hole over the nail, we then find that the plate will remain at rest in any position whatever. This peculiarity of the point P will be more readily perceived if we make a hole in the plate at a point Q near to P. When the plate is suspended from Qit will only be at rest in one position, t.e., when P is vertically beneath Q.

It must surely be regarded as a matter worthy of careful notice that any plate of any figure, regular or irregular, should contain one specific point which enjoys the unique properties which the experiments show to be possessed by the point P. This point has received a name; it is called the centre of gravity.

§ 7. Centre of Gravity of a Rigid Body.— In the illustration we have just given, we have spoken merely of a thin plate, because the experiments were more easily conducted in a body of this nature than in one of entirely irregular form. It must not, however, be supposed that a thin plate of uniform thickness is the only kind of body which possesses a point having the properties we have described. No matter what be the shape or materials of which a rigid body is composed, it possesses a centre of gravity. Let ADBC (fig. 4) be a body of any kind, and let it be suspended by a cord from a point A. Then when the body is at rest it assumes a certain position. We may suppose that a vertical hole AB is drilled through the body in the direction of the cord by which the body is suspended. If we now suspend the body by another point on its surface, C, the body will come to rest in the position which is represented in fig. 5. It will be found that these two straight holes intersect in the interior of the body at G. In fact, if we thrust a knitting needle through one of the holes, and then attempt to thrust a second knitting needle through the other hole, we shall find that the way is stopped in the interior of the body by the first knitting needle. If the body be now suspended from any other point on its surface, and if a similar hole be made through the point and in the direction of the string by which the body is suspended, it will be found that this hole also passes through the intersection of the two former holes. From each and every point of suspension the same result is obtained, and thus we are led to the conclusion that in a rigid body of any shape or materials whatever there is one point which possesses the remarkable property thus stated :—

When a body suspended by a cord from a fixed point is at rest, there is one special point which is always vertically beneath the point of suspension, whatever may be the point of the body to which the cord is attached. This point is called the centre of gravity.

In the case of a homogeneous body of regular shape, the centre of gravity is determined from the most simple considerations of symmetry. In the case of a sphere it is obvious that the centre of gravity must lie at the centre, for there is no other point symmetrically related to the figure. In the case of a parallelepiped the centre of gravity is also situated at the centre of volume. This is found by joining the opposite corners of the figure, and thus making a diagonal; joining another pair of opposite corners we have a second diagonal; and the intersection of these two lines gives the centre of gravity of the mass.

§ 8. Gravitation of a Rigid Body reduced to One Force.— A body of any description may be considered to be composed of an innumerable multitude of small particles of matter. Each of these particles is acted upon by the attraction of the earth. Each particle is therefore urged towards the earth by a certain force which tends towards the earth’s centre. The centre of the earth being nearly 4000 miles distant, the directions of these forces may, for all practical purposes, be regarded as parallel. Even if two particles were a mile distant, the inclination of the directions of the two forces is under a minute. We may therefore treat the forces as parallel without making any appreciable error. Two parallel forces may be compounded into a single force which is parallel to the two components. The forces acting on the two particles of the body may therefore be replaced by a single force. This force may be similarly compounded with the force acting on a third particle of the body, this resultant with the force on a fourth particle, and so on until all the particles have been included. We thus come to the important conclusion that the effect of gravity upon a body of any shape is to produce one force which acts vertically downwards. It remains to be shown that the direction of this force passes through the centre of gravity of the body. Suspend a body of any shape by a cord, then when the body is at rest the centre of gravity must lie vertically beneath the point of suspension. If the direction of that one force which constitutes the effect of gravity does not pass through the centre of gravity, then its line of action cannot coincide with the direction of the cord of suspension; but it is impossible that two forces should equilibrate unless their lines of action are coincident, whence we are led. to the important conclusion that the effect of the attraction of the earth upon a rigid body is to produce a single force which passes through the centre of gravity of the body and acts vertically downwards.

III. Universal Gravitation.

§ 9. Law of Gravitation between Two Masses.— Investigations of the motions of the planetshave conducted us to the conclusion that each planet is attracted towards the sun by a force which varies according to the inverse square of the distance. We likewise see that each of the satellites appears to be attracted towards the corresponding primary planet by a force which obeys the same law. We are therefore tempted to generalize these results into the proposition that any two masses in the universe attract each other with a force which varies according to the inverse square of the distance. Observations of the most widely different character have combined to show us that this law, which was discovered by Sir Isaac Newton, is true. It is called the laιυ of Gravitation.

We suppose that the distance between the two bodies is so exceedingly great compared with their dimensions that we may practically consider each of the bodies as a particle.

Let m, m' be the masses of the two bodies, and let r be the distance. The force with which m attracts m' is equal in magnitude though opposite in direction to the force with which m' attracts m. The reader may perhaps feel some difficulty at first in admitting the truth of this statement. We speak so often of the effects which the attraction of the sun produces on the planets that it may seem strange to hear that each planet reacts on the sun with a force precisely equal and opposite to the force with which the sun acts upon the planets. We can, however, by the aid of a simple illustration, show that such is really the case. Suppose the sun and the earth to be placed at rest in space, and abandoned to the influence of their mutual gravitation. It is evident that the two bodies would begin moving towards each other, and would after a certain time come into collision. If, however, the earth and the sun had been separated by a rigid rod, it would then be impossible for the two bodies to move closer to each other, so we must consider whether any other motion could be produced by their mutual attraction. As the two bodies were initially at rest, it is clear that there will be no tendency of the rod to move out of the line in which it was originally placed, and consequently if the rod begin to move at all it must move in that line, and carry with it the earth at one end, ami the sun at the other. As, however, the earth and the sun would remain separated at a constant distance, the statical energy due to their separation would remain constant, and therefore if they began to move we should have the kinetic energy of their motion created out of nothing, which is now well known to be impossible. It therefore follows that, under the circumstances we have assumed, the rod would remain for ever at rest. But what are the forces which act upon the rod? At one end of the rod the earth presses upon it with a force which is equal to the attraction of the sun upon the earth. At the other end the sun presses upon the rod with a force which is equal to the attraction of the earth upon the sun. But since the rod remains at rest, these two forces must be equal and opposite, and hence the force with which the sun attracts the earth must be equal and opposite to the force with which the earth attracts the sun.

If we express the gravitation of two masses in the form fm,m')÷r i , then m and m' must enter symmetrically into the expression, for if the two masses were interchanged the gravitation must not be altered.

We should here also advert to a circumstance connected with gravitation which is of the very highest importance. Suppose we take two definite masses (for simplicity, two pounds) separated at a definite distance (for simplicity, one foot), then the gravitation of these two masses to each other is a certain definite force (which we shall subsequently calculate). What we want here to lay special stress upon is that, so far as we know at present, this force appears to be the same whatever be the material of which the two masses are composed. Thus two pounds of iron at a distance of one foot attract each other with the same force as a pound of iron would attract a pound of lead at the same distance. The significance, or perhaps it should be said the vast importance, of this statement is apt to be lost sight of from a somewhat peculiar cause. It must never be forgotten that, when it is asserted that two masses are equal to each other, what is really meant is that, if equal forces were to act upon each of the masses for the same time, the masses would receive the same velocity. As this test of the equality of masses is not practically convenient, the weighing scales have come into use for the purpose; and though it appears to be true that when two masses have equal “weights,” as tested by the scales, then the masses are themselves equal, yet this is so far from being an obvious or necessary truth that it really is the most remarkable phenomenon connected with gravitation. The expression for the gravitation of two attracting masses must therefore depend solely upon their masses, upon their distance, and upon some specific constant which is characteristic of the intensity of gravitation.

Experiment shows that the gravitation of a body towards the earth is directly proportional to its mass, and hence we see that the expression „

y(rn,τn)÷r a must be proportional to m, and as f {m, m') must be unaltered by the interchange of m and m', it appears finally that the gravitation of the two masses is emm÷r t , whence c is a numerical constant which is equal to the gravitation of two units of mass placed at the unit of distance. To form a definite conception of the intensity of this force, we take some specific instances. It can be shown that two masses A and B, each containing 415,000 tons of matter, and situated at a distance of one statute mile apart, will attract each other with a force of one pound. If the masses of A and B remain the same, and if the distance between them be increased to two miles, then the intensity of the force with which the two masses gravitate together is reduced to one quarter of a pound. If either of the masses were doubled, the distance being unaltered, then the force would be doubled. If both the masses were doubled, then the force would be quadrupled.

§ 10. Motion of a Planet round the Sun.— The effect of gravitation when the bodies are in actual motion must next receive our attention. It may so happen that in consequence of the attraction of gravitation one of the bodies will actually describe a circle around the other, so that, notwithstanding the effect of the attraction, the distance between the two bodies remains constant. We shall first explain, by elementary considerations, how it is possible for a planet to continue to revolve in a circular or nearly circular orbit about the sun in its centre; and then we shall proceed to the more exact consideration of the form of the orbit, and the laws according to which that orbit is described.

Let S represent the sun (fig. 6), and let T be the initial position of the planet. If the planet be simply released, it will immediately begin to fall along the line TS into the sun. If, on the other hand, the planet were initially projected along the line TZ perpendicular to TS, the attraction of the sun at S will deflect the planet from the line TZ which it would otherwise have followed, and compel the planet to move in a curved line. The particular form of curve which the planet will describe depends upon the initial velocity. With a small initial velocity the deflecting power of the sun will have a more speedy effect than is possible when the initial velocity is considerable. The rapidly curving path TX will therefore correspond to a small initial velocity, while the flatter curve TV may be the orbit when the initial velocity is considerable. As the movement proceeds, the velocity of the planet will generally alter. If the planet were moving along the curve TY, it is at every instant after leaving T going farther away from the sun. It is manifest that the planet is thus going against the sun’s attraction, and therefore its velocity must be diminishing. On the other hand, when the planet is moving along the curve TX, it is constantly getting nearer the sun, and the effect of the sun’s attraction is to increase the velocity. It is therefore plain that for a path somewhere between TX and TY the velocity of the planet must be unaltered by the sun’s attraction. With centre S and radius ST describe a circle, and take a point P on that circle exceedingly near to T. With a certain initial velocity it is possible to project the planet so that it shall describe the arc TP. The attraction of the sun always acts along the radius, and hence in describing the arc TP the planet has at every instant been moving perpendicularly to the sun’s attraction. It is manifest that under such circumstances the sun’s attraction cannot have altered the velocity, for it would be impossible to give a reason for the velocity having been accelerated which could not be rebutted by an equally valid reason for the velocity having been retarded. We thus see that the planet reaches P with an unchanged velocity, and at that point the direction of the motion is perpendicular to the radius. It is therefore clear that after passing P the planet will again describe a small portion of the circle, which will again be followed by another, and so on, i.e. , the planet will continue to move in a circular orbit. We have therefore shown that, if a planet were originally projected with a certain specific velocity in a direction at right angles to the radius connecting the planet and the sun, then the planet would continue for ever to describe a circle around the sun.

§ 11. On the Elliptic Motion of the Planets.— The laws of the motions of the planets were discovered by Kepler by means of calculations founded upon observations. They may be thus stated:—

1. Each planet moves in an ellipse in one focus of which the sun is situated. 2. The radius vector drawn from the sun to the planet sweeps over equal areas in equal times.

3. The squares of the periodic times of the motions of the planets round the sun are in the same ratio as the cubes of their mean distances.

These three laws form the foundations of that branch of astronomy which is called Physical Astronomy, and they are generally known as Kepler's Laws.

The sun and the planets are all very nearly spherical, and for the present, whenever we speak of the motions of the sun or the planets, we are to understand the motions of the centres of the corresponding spheres. Indeed the diameters of these spheres are so small in comparison with the distances at which they are separated from each other, that we may generally regard them as mere physical points. Thus though the diameter of the sun is ten times greater than the diameter of the greatest planet, Jupiter, yet the sun’s diameter is only the forty-second part of its distance from the nearest planet, Mercury, and it is less than the three-thousandth part of its distance from the outermost planet, Neptune.

The orbits of the planets are so little eccentric that at a first glance the majority of them appear to be circular. Among the larger planets the orbits of Mercury and of Mars have the greatest eccentricity, being about ⅜ and ⅛ respectively. Next to these comes the orbit of Saturn, which has an eccentricity of ⅛ Next in order come Jupiter, Uranus, the Earth, Neptune, and Venus, with eccentricities of ⅛ ⅛ ττs> ττκ> τ⅛κ respectively. The orbits of some of the minor planets are, however, much more eccentric.

§ 12. The Hodograph. — In discussing the actual motion of a planet around the sun, it is very convenient to introduce the elegant conception of the hodograph. The use of this curve is originally due to

Bradley, but for its practical development we are principally indebted to Sir W. R. Hamilton. Let AB (fig. 7) be a portion of the path of a particle P acted upon by any forces. From any point 0 draw radii vectores OR 1 , OR 2 , OR 3 , OR 4 (fig. 8), parallel to the tangents to the curve AB drawn at the points, P 1 , P 2 , P 3 , P 4 , and equal to the velocities at those points. Then the curve R 1 R 2 R 3 R 4 Is the hodograph of the orbit of P. To each position P of the particle in the path there is a corresponding point R in the hodograph; and simultaneously with the motion of P in its orbit we have the motion of R in the hodograph.

The utility of the hodograph depends upon the theorem that the force which acts upon the particle is at any time equal and parallel to the velocity of the corresponding point in the hodograph. This theorem is thus proved. Let P 2 and P 3 , and therefore R 2 and R 3 , be very close together. The velocity imparted to the particle in passing from P a to P 3 in the small time δt must, when compounded with the velocity OR 2 , produce the velocity OR 3 . This increment of velocity must therefore be equal and parallel to R 2 R 3 . If therefore v denote the velocity of R, we have vδt for the velocity acquired in the time δi. The force acting on P must therefore, by the second law of motion, be equal to v and parallel to the line R 2 R 3 , which ultimately coincides with the tangent. The theorem is therefore proved.

Kepler’s second law states that in the movement of a planet around the sun the radius vector from the sun to the planet describes equal areas in equal times. It can be shown that when this is the case the force which acts upon the planet must necessarily be directed towards the sun. Let the planet be supposed to move from P to P ’ (fig. 9) in the time t with the velocity v. Then, since equal areas are described in equal times, we must have the area OPP' equal to hδt, where A is a constant which represents the area described in the unit of time. Let fall OT perpendicular upon the tangent at P, then vOT = 2A. We therefore infer that the velocity of a planet is inversely proportional to the perpendicular .et fall from the centre of the sun upon the tangent to the orbit. Produce OT to Q, so that OT × OQ is constant; the radius vector OQ will therefore be proportional to the velocity; and as OQ is perpendicular to the tangent at P, it follows that the locus of Q must be simply the hodograph turned round through 90°. Draw two consecutive tangents to the orbit, and let Q 1 and Q 2 be the corresponding points (fig. 10). Then since OT 1 × OQ 1 = OT a × OQ 2 , the quadrilateral T 1 Q 1 Q a T a is inscribable in a circle, and therefore the angles OQ,Q a and OT a T 1 are equal; whence it is easily seen that Q 1 Q a is perpendicular to OP. It follows from the principle of the hodograph that the force on P must be directed along OP.

Let F (fig. 11) be the focus of the ellipse in which, according to Kepler’s first law, the planet is moving, then, from a well-known property of the ellipse, the foot of the perpendicular FT let fall from F on the tangent drawn to the ellipse at the point P lies on the circle of which the diameter is the axis major of the ellipse. The line FT cuts the circle again at Q, and as FT × FQ is constant, the radius vector FQ must be constantly proportional to the velocity, and therefore the hodograph will be the circle TAQ turned round through 90 υ . We have already shown that the tangent to the hodograph is parallel to the force. The line CQ must therefore be parallel to FP. If θ be the angle PFC, then, in consequence of the law of equal description of areas in equal times, FP 2 d0 ÷ dt must be constant, and therefore dθ÷dt must vary inversely as FP 2 ; but the velocity of Q will be proportional to dθ ÷ dt ; hence the velocity of Q, and therefore the force, will be inversely proportional to FP a . In this way it is shown that the planets are attracted by the sun. with a force which varies inversely as the square of the distance.

§13. The Problem of Three Bodies. — We have pointed out that the motion of a planet is determined by the mutual attraction which exists between the planet and the sun. It follows, however, as a necessary consequence from the law of universal gravitation, that each planet is attracted not only by the sun but by every other planet, and indeed, so far as we know, by every other body in the universe. The effect of the attraction of the sun preponderates so enormously over the other attractions that the obedience of the planets to Kepler’s laws, which would be perfect were the sources of disturbance absent, is still so nearly perfect that the departure therefrom can only be perceived either by very accurate or by very long-continued observations. The refinements of modern observations have, however, brought to light a very large number of perturbations, as they are called, in the motions of the planets and their satellites, which are due to the interference of other bodies. The explanation of these different perturbations by the law of universal gravitation has proved in the great majority of cases triumphantly successful.

In any problem where perturbations are involved, we have at least three bodies, viz., the principal body S, a body P which circulates around S, and a disturbing body P'. If we had only the two bodies S and P to consider, then P would describe around S a conic section, of which S was the focus, and the radius vector SP would sweep over equal areas in equal times. By the introduction of the third body P', which attracts both the former bodies S and P, the motion is deranged, the orbit which P describes is no longer a conic section, and its radius vector has ceased to describe equal areas in equal times. The general case of this problem, as we have here described it, is one of excessive complexity. It is, however, a fortunate circumstance that, up to the present time, astronomers have had but little occasion to attack the problem in its general form. 1 In the solar system we have a great number cf different problems of perturbations to discuss, but there is one feature common to all these problems. This feature is that the perturbing force is very small in comparison with the primary force, by which the motion of the disturbed body is controlled. In consequence of this peculiarity, the problems of perturbations in the solar system become greatly simplified. The circumstances of the orbits of the planets and their satellites are also happily such as to furnish additional facilities in the solution of the problems of perturbations. The eccentricities of the orbits are small, and in nearly every case the inclinations are small also. The problems of perturbations in the solar system are consequently adapted for the methods of successive approximation; but we should perhaps warn the reader that, even with the assistance so fortunately rendered by the circumstances of the case, the problems are still among the most difficult to which analysis has ever been applied.

Let S, P, P' (fig. 12) denote the three bodies of which the masses are M, m, m'. Let the distances SP, SP', PP' be denoted by r, r , , p respectively; then if e denotes, as before, the gravitation between two units of mass separated by the unit of distance, we have on S the forces and along SP and SP' respectively; on P the forces and f 1 lHL along PS and PP' respectively; on P'the forces and e 7 -ζ l along P'S and P'P respectively.

In the problem of the perturbation of a planet which is moving around the sun, the absolute motion of the planet in space is not what we are concerned with; what we do require is the relative motion of the planet with respect to the sun.

Suppose that we apply to each unit of mass of the three bodies, forces equal to ernr* 2 parallel to PS and ιm'r f ~ i parallel to P'S, then it is clear that, from the second law of motion, each unit of mass of the system will receive in a small time, so far as these forces are concerned, equal and parallel velocities, and therefore the relative motions of the three bodies will not be altered by the introduction of these forces.

But these forces will, so far as the body S is concerned, amount to eMrn÷r' 2 and ⅛'÷r' 2 in the directions PS and P'S respectively, and will therefore neutralize the attractions of P and r upon S. It follows that, so far as the relative motions only are concerned, we may consider the body S as fixed, while the body P is acted upon by the forces effi(M + ffl)÷r 2 on PS emm ’ ÷ρ i on PP' emm'÷V i parallel to P'S, with of course similar expressions for the forces on P'.

The problem of perturbation is therefore reduced to the determination of the motion of P in obedience to these forces, while S regarded as fixed.

It is plain that, in order to find the motion of P, we should know the motion of P ’, which is itself disturbed by P; but as the effect of the perturbation is small, we may, without appreciable error, use the place of P', derived on the supposition that the motion of P' is undisturbed, for the purpose of calculating disturbances caused by P' in the motion of P. We shall therefore suppose that the orbit of P' is an ellipse which is described around the focus S in consequence of an attractive force em'(M + 7n')÷r' 2 directed towards S.

In computing the disturbed motion of P we transform the three given forces into three equivalent forces, which will be more convenient for our purpose. We take first a force T directed along the radius vector; secondly, a force V perpendicular to the radius vector, and lying in the plane of the orbit; and, thirdly, a force W, which is normal to the plane of the orbit. T is to be regarded as positive when it endeavours to increase the distance between P and S; V is to be regarded as positive when it tends to move P to the same side of the line PS as that in which the direction of its motion tends to carry it; and W is to be regarded as positive when it tends to raise P to the north of the plane of its orbit.

From P' (fig. 13) let fall a perpendicular P'Q upon the plane of the undisturbed orbit cf P, and let fall from Q a perpendicular QR upon the line SP. The force along PP' can be resolved into a component parallel to QP' and another parallel to QP; and the latter can again be decomposed into components parallel to RQ and RP. Hence we have, as the equivalents to the force on PP', the following three forces which form parts of T, V, W respectively :—

+ i≡PR 5 -≡J⅛Rj √-≡'qP'.

In a similar manner we find for the components of the force parallel to SP, the following three forces which form parts of T, V, W respectively:—

- √ 3 SR; +ψ3-QRi "-^^ qγ ∙

Hence we deduce the following expressions :—

∣p isss _ ewt(M + wt) _ emm ’ pR _ !^ 7 !⅛R r s p 3 r'' i

V- -5≡.QR + ^' qr .

W- + ^QP'-^≡Qp i

Let the line SO, from which the longitudes are reckoned, be drawn in the plane of the undisturbed orbit of P. Let the longitudes of P and P' be l, l' respectively, and let the angle P'SQ or the latitude of P' be denoted by b', then we have

QP'=r' sin b' ; SQ=r' cos b'.

RQ = SQ sin (Z' - Z)=r' cos b , sin (l , - i). SR = SQ cos (l -l) = V cos b' cos (Z' - l}. PR = r - r' cos b' cos (Z' - l). 1 There can be little doubt that in the case of multiple stars the problem of three or more bodies would often have to be faced in all its ruggedness, but the observations which would be required are certainly wanting, and probably unattainable.

With these substitutions, we have for the values of T, V, W, the following expressions :—

T = - + f mm'( Ï - ⅛ V cos b' cos (l'-l)- ^r. r ∙2 ∖ p i r KJ f ^V≈ - f mm'( -A - _A )√ cos b' sin (l - l'). ∖ p i r V w = ∞ ’ (^--⅛y siπy ∙

It will be noticed that the first term of the expression for T consists of the force which corresponds to the purely elliptic motion. The two remaining terms in T, as well as the whole of V and W, depend upon the disturbing force, as is evident from the circumstance that they contain m , as a factor, and would vanish if m ’ were equal to zero. As b' is small, we may neglect its squares and higher powers, so that cos δ'=l and sin δ'=⅛'.

When these substitutions are made, we see that the expressions for T and V are both independent of b' ; and consequently we may, so far as these forces are concerned, consider the motions of the disturbing and the disturbed body to take place in the same plane.

The expression for W contains, however, the first power of the latitude of the disturbing body. This is of course connected with the circumstance that it is only in consequence of the disturbing force W that the disturbed body is induced to leave the plane of its undisturbed motion at all.

§14. Calculation of Disturbed Motion.— Observation has shown that, notwithstanding the perturbations, the orbit of each planet differs but little from a circle, of which the sun is the centre. It is further shown by observation that, though the rate at which a planet moves in its orbit is not quite constant, it is still very nearly so. We may make a similar statement with reference to the motion of the moon around the earth. The orbit of the moon is nearly a circle, of which the earth is the centre, and the velocity of the moon in its orbit is nearly constant. These features of the motions of the planets and the moon enable us to replace the more exact formulae by approximate expressions which are much more convenient, while still sufficiently correct.

Let p, θ be the polar co-ordinates of a celestial body which moves nearly uniformly in a nearly circular orbit. The form of the orbit may be expressed by an equation of the type

∕(p,θ) = O∙

It will, however, be more convenient to employ two equations, by means of which the coordinates are each expressed directly in terms of the time. We thus write two equations of the form

Ρ = /ι(0> > and by elimination of t from these equations the ordinary equation in polar coordinates is ascertained. With reference to the forms of the functions f i and ∕ 2 , we shall make an assumption.

Letχ 1 , χ 2 , &c., be arbitrary angles; w 1 , <a. 2 , &c., arbitrary angular velocities; α 1 , α 2 , &c., small arbitrary linear magnitudes; and η 2 , τ 2 , &c., small numerical factors, a is an arbitrary linear magnitude, and ω is an arbitrary angular velocity. We shall assume that p = α + α 1 cos (ω 1 i + χ 1 ) + α 2 cos (ω√ + χ a ) +, &c · θ = a>t + η l sin (ω 1 i + χ 1 ) + η i sin (a> 2 i + χ a ) +, &c.

To justify our employment of these equations it would really be sufficient for us to state that, as a matter of fact, the motions of all the heavenly bodies which are at present under consideration are capable of being expressed in the forms we have written. It may, however, facilitate the reader in admitting the legitimacy of this assumption, if we point out how exceedingly plausible are the a priori arguments which can be adduced in its favour.

The orbits of the celestial bodies are approximately circular, and consequently ρ must remain approximately constant. The value of p is, in fact, incessantly fluctuating between certain narrow limits which it does not transcend. Thus p is what is called a periodic function of the time. It is necessary that the mode in which the time t enters into the expression for p must fulfil the condition of confining p within narrow limits, notwithstanding the indefinitely great increase of which t is susceptible. It is obvious that this condition is fulfilled in the form we have assumed for ρ. Under all circumstances p = ~j>a -p a 2 -ψ- <z.-, q~, &c.; P=<α±α 1 ±α 2 ±, &c.; and as α 1 , α 2 , &c., are small quantities it appears that p is necessarily restricted to narrow limits.

By similar reasoning we can justify the equation for θ, for we can show that the angular velocity of the celestial body to which it refers must be approximately uniform.

By differentiation, dθ — = « + η 1 ω l COS (ω√ + χ 1 ) + ιj 2 ft∙ 2 cos (<√ + X 2 ) ÷ > &c ·

Under all circumstances we must have

= &c.; at

= ∙≤ω±η 1 ω. ±η 2 co 2 ±, & c ·> and as η 1 , η 2 , &c., are all small quantities, it is obvious that ~ ig confined within narrow limits, notwithstanding the indefinite augmentation of the time.

In addition to the reasons already adduced in justification of the expressions of p and θ, it is to be remarked that the number of disposable constants α 1 , α 2 , &c., χ 1 , χ 2 , &c., y 1 , y. 2 , &c., ω 1 , ω 2 , &c., a, ω is practically indefinite, and that consequently the equations can be compelled to exhibit faithfully the peculiarities of any approximately circular orbit described by a particle moving with approximate uniformity.

§ 15. Determination of the Forces. — The orbit which is described by a planet or other celestial body being given by the equations p = α+2α 1 cos (ω√ + χ i ), θ-ωt + '2,τη 1 sin (ω 1 i + χ 1 ), it is a determinate problem to ascertain the forces by which the motion of the planet is controlled. In making this calculation we shall assume that the squares and higher powers, and also the products of the small quantities a 1 , a 2 , &c., ∙η li ∙η. 2 , &c., may be discarded. T is to be computed from the well-known formula rp ∕<Z 2 p <Z0 2 ∖

τ - m u~⅛) where we have

⅞=-2α-ω l sin (a√ + χ 1 )j at

^ 2α 1 <e 1 2 cos (<√+χ 1 ); dø

— ≈ω÷2η 1 o> 1 COS (w√ + ×ι);

—- = ω 2 + 22η i α>ω 1 cos (w 1 i + χ 1 ); d0 2 Pdp =aω 2 + 2(2τj 1 αωω 1 + a 1 cβ 2 ) cos (a> l t + χ 1 ); whence by substitution

T — - wwiω 2 - 2wr(a 1 ω 1 2 + 2τ∕ 1 aωω 1 ÷ a 1 <⅛> 2 ) COS (ω 1 i + χ 1 ).

To compute V, the force perpendicular to the radius vector, we proceed as follows :—'

p 2 =a 2 + 22αα 1 cos {o> 1 t + χ 1 );

— = ω + 2η 1 ω 1 cos (ω 1 < + χι); whence we find p 2 ^ = α 2 w + 2(2im 1 ω + τ∕ 1 α 2 w 1 ) cos (ω 1 ∕ + χ 1 ). dt

Differentiating we have -j^p 2 ^j= -2(2αα 1 ωω 1 + η 1 α 2 α>ι 2 ) sin (<* 1 t + χ 1 )∙ Substituting this in the ordinary formula

V=— A ( ∖

p dt ∖ fdt) we have finally

V = — 2∕n(2α 1 ωw 1 + 17 l αω l 2 ) sin (w 1 i + Xι)∙

Let us assume, for the sake of brevity,

λ 1 = <√ + χ 1 J λ 2 = w√ + χ 2 5 &C.

αF 1 = α 1 (ω 2 + ω 1 2 ) + 2η 1 αωcu 1 ; flG 1 = 2α 1 wω 1 + t∣ 1 a⅛> 1 2 ; aF 2 = a. 2 (a> 2 + ω 2 2 ) + 2η. 2 aωo> 2 ; aG. 2 = 2a 2 ωω 2 + η 2 aω 2 2 ;

&c. &c.

then the result to which wehave been conducted maybe thus stated:— If a body be moving in a nearly circular orbit under the infl∙. ’.3nce of a radial force equal to - rnaω 2 - ma2F 1 cos λ 1 and a force perpendicular to the radius vector equal to

- wια2G 1 sin λ 1 , then the path which the body describes is defined by the equations r=α + 2α 1 cos λ 1 j 0 = ωi + 2η 1 sin λj.

§ 16. Deductions from these Expressions.— We proceed to point out a few of the more remarkable deductions from this theorem.

If as a first approximation we neglect entirely the small quantities α 1 , a 2 , &c., 1j 1 , w 2 , &c., we have r=α∙ θ = ωt∙, Υ=-maω~∙, V = 0.

In this case the orbit described by the body is a circle of which the radius is a. The angle made by the radius vector to the particle with a fixed axis is proportional to the time. It follows that the velocity with which the particle moves in its orbit is uniform. Since the angle swept out by the particle in one unit of time is equal to ω, it follows that the angular velocity of the particle is equal to ω ; since the radius is α, the actual velocity with which the particle is moving in its orbit is ωa. In the case now under consideration the force V perpendicular to the radius vector is constantly equal to zero. Hence the total force which acts upon the particle is always directed along the radius vector, and is equal to - maω 2 . The sign - merely expresses that the force which acts upon the particle must be constantly directed towards the centre.

Let p denote the periodic time of the motion of the particle in its orbit. Then we must find the angular velocity by dividing the angle 2r described in the time p by the time p, whence

2ιr — = ω. P

By substituting this value for ω in the expression for T, we deduce τ -^≡p~∙

This proves that the force must vary directly as the radius of the orbit of the particle and inversely as the square of the periodic time in which the orbit is described.

Let us now consider the case in which a 1 and η 1 are retained, while the remaining quantities α 2 , α 3 , &c., η 2 , η 3 , &c., are all equal to zero. The formulae then are r=α + α 1 cosλ 1 j T = - maω 2 - maF 1 cos λ 1 ; θ = o>t + η 1 sin λ 1 ; V = - maG 1 sin λ 1 ; where αF 1 = α 1 (α> 2 + ω 1 2 ) + 2η 1 aωω 1 ; aG 1 = 2α 1 ωw 1 + aη 1 ω 1 2 .

It is clear in the first place that the orbit is not a circle, for as λ 1 depends upon t, it will follow that cos λ 1 may vary between the limits +1 and -1, so that the radius vector r may also vary between the extreme values a + a 1 and a-a 1 . As a 1 is extremely small, it appears that the orbit is still nearly circular with a radius a, but that the particle may sometimes be found at a distance a 1 on the inside or outside of the circle.

We shall similarly find that the angular velocity with which the radius vector sweeps round is not quite uniform. The average angular velocity is no doubt ω, but the actual position of the radius vector differs by the quantity ∙η 1 sin λ 1 from what it would have been had its motion been uniform. As η 1 sin λ 1 must vary between the limits + η 1 and - η 1 , it follows that the radius vector can never be at an angle greater than η 1 from its mean place, i.e., the place which it would have occupied had it continued to move uniformly.

The orbit is completely defined by the two equations—

θ=ωt + a 1 sin (ω 1 Z + χ 1 ); r≈a + a∙η 1 cos (0qZ + χ 1 ).

If from these two equations the time t could be eliminated, the result would be the equation in polar coordinates of the path which the particle described. Owing, however, to the fact that the quantity t occurs separately, and also under the form of a sine and cosine, this elimination would be transcendental.

If, however, we take advantage of the smallness of ∕ 1 and g 1 , we can eliminate t with sufficient accuracy for all practical purposes.

As θ is nearly equal to ωt, we may assume for t as the first approximation the value θ÷ω. If we consider the squares or higher powers of α 1 and ∙η 1 negligible, this value of t may be substituted for t under the sine and cosine, and we have r=a + η l cos (- 1 β + χ 1 )

This equation in general denotes a curve undulating about the circumference of the circle of which the radius is a.

We must now briefly consider the case where the motion of the particle is not confined to a plane. Suppose a third axis OC be drawn through the point 0 perpendicular to the plane which contains the undisturbed orbit. Let z be the coordinate of the particle parallel to the line OC which is called the axis of z, while x, y denote as usual the coordinates referred to two other rectangular axes. We shall suppose that the motion of the particle is such that the coordinate z can be expressed by the equation z=A 1 sin 0ι + ⅛ 2 sin jS 2 + , &c., where h 1 , h 2 , kc., are lines of constant length; β 1 , β 2 , &c., are angles of the form p√ + ξ 1 , p 2 t + ξ 2 , &c.; p 1 , p 2 , &c., ξ 1 , ξ 2 , &c., are constants; and t denotes the time.

Let W denote the force which acts upon the particle P in a direction parallel to the axis of z. Then we have w -⅛∙

Differentiating the equation z*∙h 1 sin β 1 + h 2 sin j8 2 + , &c., we have Tt =hi cos 3ι ¾ + h * cos 3a ¾ + > &c ·; tat l*-Λ' ⅞ , -⅛ s ≈∙> whence ~ =Pιhι cos β 1 +p 2 h 2 cos β 2 +, &c.

Differentiating again d 2 s — = -p 1 2 h 1 sin p l - pfh 2 sin β 2 - . &c; whence, finally,

W = -rnp 1 2 h 1 sin β 1 - rnp 2 2 h 2 sin β 2 - , &c.

§ 17. The Motion of the Moon.— One of the most important problems to which we may apply the expressions to which we have been conducted is to an examination of the disturbances which the moon experiences in its motion round the earth. The moon would describe a purely elliptic motion around the earth in one of the foci were it not that the presence of the sun disturbs the motion and produces certain irregularities. Notwithstanding the vast mass of the sun, these disturbing causes still only slightly derange the moon’s motion from what it would be were these disturbing causes absent. The reason of this is that the sun is about 400 times as far from the earth as the moon, and consequently the difference of the effects of the sun upon the earth and the moon is comparatively small.

In applying the formulae to the case of the moon we denote by

S, P, P' the earth, the moon, and the sun respectively;

M, wι, m' signify the masses of the earth, moon, and sun; r, r ’ are the distances of the moon and the sun from the earth; ρ is the distance from the moon to the sun; l, l' are the longitudes of the moon and the sun measured in the moon’s orbit.

T, V, W are the forces acting upon the moon, whereof T is along the radius vector, V is perpendicular to the radius vector, and W is perpendicular to the plane of the moon’s orbit.

Since r÷r l is very nearly equal to l÷400, we may regard this fraction as so small that its squares and higher powers are negligible. Hence, since cos Z>'=l, p = 7∙'^1-2^cos (Z-Z') + =√(1-L cos (Z-Z')); whence p~ 8 =r'- 3 ( 1 + Si cos (Z - Z ’)J,

» ∣-⅜-≡⅛'∞(≈-n∙

With these substitutions we have, after a few simple transformations, T- - t " ι (M÷ ¾∣ i~⅛(l+8 ∞> 2(Z-θ) i v--∣ ≡2ista2(z-r)i W-3 e ^T cos (Z-Γ) sin 5'.

§ 18. The Variation.— The inequality in the motion of the moon which is known as the variation is independent both of the eccentricity of the orbit of the moon and of that of the earth. As we shall at present only discuss irregularities in longitude and radius vector, and as we shall neglect small quantities of an order higher than the second, we may assume that the plane of the orbit of the sun coincides with the plane of the orbit of the moon.

The radius vector and the longitude may be expressed by the formulae r=α + q∕ ’ 1 cos λ 1 ; Z= ωt + g 1 sin λ 1 .

We shall assume that

2(Z ’-Z)=λ 1 j and since r^ 2 =α~ s (l - 2∕ 1 cos λ 1 ), we have rp _ em(M + m) j_ 1 (mm ’ a α 2 2 α' 3

, ∕ n , rn0n + M) , 3 trnm'a ∖ „„„ . .

^ l V 1 a> + 2 ^ιH∞ sλ > i but from § 15, n being the mean motion of the moon,

T = - mn 2 a - rnaF 1 cos λ 1 ; whence, by identifying the two expressions we have from the portions independent of the time,

⅛, <M + m) J ern ’ α α 2 2 α' 3

This is a very important formula, inasmuch as it gives the relation between the mean motion n and the mean distance a in the disturbed orbit.

Suppose that there were no disturbing influence, and that a satellite moved uniformly around the earth in a circular orbit of radius α 0 with a mean motion n 0 , then we have e(M+wι)-n 0 ’ α 0 8 .

Assuming also that the earth moves uniformly round the sun in a circular orbit of radius a' with a mean motion w, then e(M+τn')-n' 2 α' 3 , or, since M is negligible in comparison with m', on'=-nV.

With these substitutions, and making w≈ri 3 ÷n 2 , we have finally ri 2 a 3 =n 0 2 a 0 3 (l - 1 in).

If we identify the coefficients of cos λ 1 in the two expressions for T, we have

-≡>F 1 -% ’ 2¾t=>+∣i=⅛; whence, by substitution and by neglecting small quantities, ∕ 1 (n 2 + n 1 2 ) 4- 2ft≡ 1 - - 2∕ 1 n 2 - ∣wπ 2 .

We have also for V the two expressions

3 tmvι'r . ,

~2~7*~ 8ΐηλ1 ’

and -τnαG 1 sinλ 1 j whence by identifying the coefficients

2∕ 1 ≡ 1 + g 1 n 1 2 = ¾wn 2 , where n 1 denotes the rate at which λ 1 alters.

Solving the two equations, we deduce the following expressions :— f 3w n⅛ 1 + 2n 3 . _ 3w n 2 (n 1 2 + 2nn 1 + 3π 2 )

71=> ~T n 1 3 -w 2 n 1 ’ 71 2 n 1 2 (V→ 2 )

We can now calculate the value of f x and g x numerically. For ¾ ≈ 2 to ~ n ' ≈ ⅞ 365'256- 27’322 -1 .gκθ 1 η n 365'256 ’

and also w = -=——;

178∙72 ’ from these we deduce

∕ 1 = -0 007204

<Z 1 =≈+0O1024 = 35'∙20.

The results at which we have arrived may be thus summarily stated, using, however, a more accurate value of g 1 than that which is formed by this method :—

If we suppose the orbit of the moon to coincide with the plane of the ecliptic, and if we neglect the ellipticity of the orbit of the earth around the sun and that of the moon around the earth; if n' denotes the mean motion of the earth around the sun, and n the mean motion of the moon around the earth; and, finally, if n and a be connected by the equation ii 2 α 3 = 0 - 9972 ∈(M+m)

then we have for the motion of the moon the equations r=a(l -0 ’ 007204 cos (2nt-2n't)^, l=nt + 39'30" sin (2nt - 2n ’ t).

We thus see that the motion of the moon, on the hypothesis which we have assumed, is different from a uniform circular motion. The distance from the moon to the earth is sometimes l-139th part greater or less than its mean value. And the longitude of the moon is sometimes 39' 30" in advance of or behind what it would be on the supposition that the moon was moving uniformly.

When the distance is the greatest, we have cos (2nt - 2n't) = -1; whencβ 2rf-2Λ-.or3x i and it follows that the distance of the moon from the earth is greatest at quadratures. When the distance is least, then cos (2nt -2rit)≈ +1, whence 2nt- 2n't=Q or 2π; consequently the moon is nearest to the earth at syzygy. It thus appears that the orbit of the moon as modified by the disturbing influence of the variation resembles an oval of which the earth is the centre, and of which the minor axis is constantly in the line of syzygies.

The mean longitude of the moon and the true longitude coincide when sin {2nt- 2n't) = 0.

This condition is fulfilled both at syzygy and quadrature; consequently the mean place of the moon and its true place coincide when the moon is either in syzygy or in quadrature. The true place of the moon is at its greatest distance in advance of the mean place when sin (2πi- 2n'i) = l.

This condition is fulfilled at the middle points of the first and third quadrants, while at the middle points of the second and fourth quadrants sin (2nt ~ 2n ’ t) = -1; and therefore the moon is behind its mean place in the second and fourth quadrants.

After new moon, the distance between the moon and the earth gradually increases, and the apparent velocity of the moon also increases until, when the moon is three or four days old, it has advanced 39' beyond its mean place; the velocity then begins to diminish, though the distance goes on increasing, until at first quarter the distance has attained a maximum. After first quarter the distance diminishes, and the moon falls behind its mean place, the maximum distance of 39' behind the mean place being reached about 11 days after new moon. At full moon the distance has become a minimum, and the’ mean place and true place coincide. At 18 days the true place has again gained 39' on the mean place, but the distance increases, and at third quarter the distance is again a maximum, and the true and the mean place coincide. After passing third quarter the distance diminishes, and the true place falls behind the mean place, the difference attaining a maximum on the 26th day, after which the true place gains on the mean place, with which it coincides at new moon, when also the distance is again a minimum.

Since the amount of this irregularity in longitude is so considerable, being in fact larger than the diameter of the moon itself, it is very appreciable even in comparatively coarse observations. It was discovered by observation by Tycho Brahe, by whom it was named the variation.

It would lead us too far to endeavour to trace out any of the other irregularities by which the motion of the moon is deranged. We have taken the variation merely as an illustration of one of the numerous corrections which the law of gravitation has explained. The accordance which subsists between the values of these corrections as computed by theory and as determined by observation affords the most conclusive evidence of the truth of the law of universal gravitation.