through 59,400 feet in one minute near the earth's surface; hence, as 602: 12; or as 3,600:1:: 59,400 feet: 167 feet, which is the space through which a body would fall in a minute at the distance of the moon. Now this agrees, making allowance for using round numbers, with the actual distance through which the moon would fall if the centrifugal force were to cease. The moon therefore is retained in her orbit by gravity, and gravity only, for it would be unphilosophical to assign two causes to account for effects precisely similar.

§ 364. Thus also the ratio of the forces of gravitation of the moon towards the sun and the earth may be estimated. For, 365 days being the periodic time of the earth and moon about the sun, and 27.3 days the periodic time of the moon about the earth; also 60 being the distance of the moon from the earth in terms of the earth's radius, and 23,920 her mean distance from the sun in the same 23920 60 measure, we have 365.252 27.32

:: F:f:: 2 : 1 near

ly; that is, the moon's gravitation towards the sun is to her gravitation towards the earth as 2 to 1 nearly.

Again, from the same principles, the centrifugal force of a body at the equator, arising from the rotation of the earth, is derived. For these propositions apply to centrifugal forces as well as centripetal ones, the terms, being correlatives (where these two alone keep the body in its orbit). And we have just found that the time of revolution is 5,075s. when the centrifugal force becomes equal to the gravity; also it appears that the forces in circles having the same radii are reciprocally as the squares of the periodic times; hence, therefore, since the earth's rotation is performed in 23h. 56m., or 86,160s., we have 86,1602 5,0752: the force of gravity: the centrifugal force of a body at the equator arising from the earth's rotation :: 1: nearly.

§365. Since the time of revolution of a body under the equator, and in any parallel of latitude, is equal; the centrifugal forces are as the distances from the axis of motion, or, as the radius to the cosine of the latitude. But in any latitude the centrifugal force is not (as under the

equator) opposite to the whole gravity, but only a part of it; which also is to the whole as the cosine of the latitude to radius.

Therefore combining these two ratios, it follows, that the diminution of gravity at the surface of the earth, arising from the centrifugal force, varies as the square of the cosine of the latitude.

The law just stated for the diminution of gravity is on the supposition of the earth's sphericity; but as the polar axis of the earth is rather shorter than the equatorial, the former being to the latter nearly as 300 to 301, or what is technically denominated the compression being about 300, the preceding theory is not exact.

§ 366. The pendulum serves as an excellent measure of the force of gravity, for by it may be ascertained the distance through which a body unsupported would fall in a second of time. The following proportion will always give that space.

As 3.1416: 1 :: 1 second: the time of falling through a space equal to half the length of a pendulum beating seconds.

But the spaces described by falling bodies are in the proportion of the squares of the times; therefore:

As the square of the time last found is to one second squared, so is half the length of the pendulum beating seconds to the space through which a body would fall in a second; but this is the measure of the force exerted by the attraction of gravity.

For example, suppose it is found by observation that the length of a pendulum beating seconds, in the latitude of London, is 39.126 inches, or 3.2605 feet; required the space through which a body would fall in a second in that latitude.

As 3.1416: 1 :: 1 sec.: 3183 sec., the time a body would take to fall through 1.63025 feet, half the length of the pendulum beating seconds.

Again: As 31832, or 10131489 sec. : 12 sec. :: 1.63025 feet, one half the length of such a pendulum, in latitude 51: 16.09 feet, the space through which a body would fall in a second, in that latitude.

§ 367. Suppose the shorter axis of an ellipse to diminish continually, the longer axis remaining the same; the ellipse will be transformed into a straight line, equal in length to the major axis. In all the successive ellipses produced by this gradual diminution of the minor axis, the periodic time remains unchanged, if the force acting at the centre remains unchanged. The ellipse may be considered as undistinguishable from the major axis, and the revolution in such an ellipse as undistinguishable from the ascent of a body along the axis, to its subsequent descent in an equal time. Consequently a body solicited by such a central force will descend through the space in half the time of revolution in the ellipse. Let T be the time of revolution of a planet at any distance, and t the time of a revolution at half that distance; then, by the third law of Kepler, T2: t2 :: 23 : 13; hence, t =

Τ

and t

T

/32

;

but we have just seen that it is the time in which a body would fall to the sun from the distance corresponding to T; therefore, the time in which a planet would fall to the sun by the action of the centripetal force is equal to its periodic time divided by 32; or it is equal to that time multiplied by the reciprocal of 32, that is, by 0, 176776. By this general rule, the times in which the different planets would reach the sun, if let fall when at their mean distances, may be determined.

CHAPTER XVI.

PERTURBATIONS.*

Disturbing Forces. Problem of the three bodies. Stability of the Solar System. Periodical and Secular Inequalities. Perturbations in Longitude. Motion of the Line of Apsides. Variation of the Eccentricities. Perturbations in Latitude. Retrogradation of the Nodes. Variation of the Inclinations. Permanency of the Major Axes. Effect of a Resisting Medium. Invariable Plane of the Solar System. Inequality in the Theory of Jupiter and Saturn.

§ 368. We must now introduce some modification into the facts and the laws we have been asserting. Since all members of the solar system are exposed to one another's influence, and are free to move, they cannot retain the motions which the sun's influence alone would · impress on them. Were the planets attracted by the sun only, they would describe perfect ellipses; as each planet and satellite attracts every other planet and satellite, they move in no known or symmetrical curve, but in paths now approaching to, now receding from the elliptical form. Thus we find in the heavens no perfect ellipse, no immovable plane, no unvarying motion; no cubes of the distances bear precisely the same proportion to the squares of the times, no radius vector sweeps over equal areas in exactly equal times. The areas really described, however, the departures from an elliptic path, the alterations of the planes, and the motions of the nodes and the apsides become the tests of the disturbing forces.

An attraction which acts equally and in the same direction on two bodies does not disturb their relative motions. The force which disturbs the motion of a satellite or a planet is the difference of the forces which act on the central and revolving body. Thus if the moon is between the sun and the earth, and if the sun's attraction, in a certain time,

*This chapter is taken from Mrs. Somerville's "Connection of the Physical Sciences."

draws the earth 200 inches, and in the same time draws the moon 201 inches, then the real disturbing force is the force which would produce in the moon a motion of one inch from the earth. If the direction of the attracting force is different in the two cases, some complication is introduced; but by the resolution of forces the amount of disturbance in any given direction may be found.

The disturbing body may be exterior to the orbit of the revolving body, as Jupiter to the earth; or within it, as Venus to the earth; or it may be central and fixed, as the sun, while the two bodies whose relative motions are disturbed both revolve round it.

$369. The simplest mode of considering perturbations is however to consider merely the amount of force and the direction in which it is exerted, without regard to the body exerting it.

To determine the motion of each body, when disturbed by all the rest, is beyond the power of analysis. It is therefore necessary to estimate the disturbing action of one planet at a time, whence the celebrated problem of the three bodies, originally applied to the moon, the earth, and the sun; namely, the masses being given of three bodies projected from three given points, with velocities given, both in quantity and direction; and, supposing the bodies to gravitate to one another with forces that are directly as their masses, and inversely as the squares of their distances, to find the lines described by these bodies, and their positions at any given instant or in other words, to determine the path of a celestial body when attracted by a second body, and disturbed in its motions round the second body by a third-a problem equally applicable to planets, satellites, and comets.

By this problem the motions of translation of the celestial bodies are determined. It is an extremely difficult one, and would be infinitely more so, if the disturbing action were not very small when compared with the central force; that is, if the action of the planets on one another were not very small when compared with that of the sun. As the disturbing influence of each body may be found separately, it is assumed that the action of the whole sys