form. Since the angle swept out by the particle in one unit of time is equal to <o, it follows that the angular velocity of the particle is equal to o> ; since the radius is a, the actual velocity with which the particle is moving in its orbit is u>a. In the case now under consideration the force V perpendicular to the radius vector is con stantly equal to zero. Hence the total force which acts upon the particle is always directed along the radius vector, and is equal to - maa>*. 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 2ir described in the time p by the time p, whence 2ir - = 60. P By substituting this value for o> in the expression for T, we deduce 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 l and 7^ are retained, while the remaining quantities a z , 3 , &c., rj.,, 7j 3 , &c., are all equal to zero. The formula; then are r a + a 1 cos A x ; T = - wia&> 2 - ?naF x cos AJ ; 6 =ut + ri 1 sin AJ ; V= - maG 1 sin A : ; where aF 1 = re 1 (co 2 + a> 1 2 ) + 2r; 1 rcw 1 ; Gj = 2 1 a>co 1 + aijjo^ 2 . It is clear in the first place that the orbit is not a circle, for as AJ depends upon t, it will follow that cos A. x may vary between the limits + 1 and - 1 , so that the radius vector r may also vary between the extreme values a + a l and a - a^ 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^ 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 u, but the actual position of the radius vector differs by the quantity r} 1 sin Aj from what it would have been had its motion been uniform. As r)! sin AJ must vary between the limits + n l ftnd - rjj, it follows that the radius vector can never be at an angle greater than j] l 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 l sin (ea 1 t cos 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 / t and g v we can eliminate t with sufficient accuracy for all practical purposes. As is nearly equal to tat, we may assume for t as the first approximation the value 0-j-. If we consider the squares or higher powers of a v and rjj negligible, this value of t may be substi tuted for t under the sine and cosine, and we have 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 perpendicular to the plane which con tains 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 2 = Aj sin 0J + &2 sin 2 + , &c., where h lt Ti z , &c., are lines of constant length ; j8 I; &. 2 , &c. , are angles of the form p ] t + ^ l , ^ + 2 > &c - 5 Pi, Pz, &c -> L tu &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 Differentiating the equation 2 = Aj sin #1 - we have _ = Ji, cos B, a but whence dz -,- ~~TJ ^ Pz ^ * 5 COS /Sj + ^g&jj COS Differentiating again d 2 ~ 4P~ ~ whence, finally, Pi ~ Sil1 02 - : &C , &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 pro duces 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 formula to the case of the moon we denote by S, P, P the earth, the moon, and the sun respectively ; M, m, m signify the masses of the earth, moon, and sun; r, r are the distances of the moon and the sun from the earth ; p is the distance from the moon to the sun ; Z, Z 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 1 is very nearly equal to l-f-400, we may regard this frac tion as so small that its squares and higher powers are negligible. Hence, since cos & = whence or JL- * =3 ,cos(Z-Z )- p.i ? . d r i With these substitutions we have, after a few simple transformations, T= - - - + - - ,. ( 1 + 3 cos 2(Z- Z ) r 2 2 r 3 V w = 3 ??i?,V cog ^_ ^ gin ^ r 3
§ 18. The Variation.—The inequality in the motion of the moon which is known as the variation is independent both of the eccen tricity 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 formula? r = a + a/j cos 1 ; l = (at + g^ sin A r We shall assume that 2(Z -Z) = A l5 and since we have + 2 but from 15, n being the mean motion of the moon, T = - mifia - maF x cos A x ; whence, by identifying the two expressions we have from the portions independent of the time, m) _ 1 em a ~" ~~ This is a very important formula, inasmuch as it gives the re lation 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 a with a mean motion n , then we have
[ equation.]
Assuming also that the earth moves uniformly round the sun in a circular orbit of radius a with a mean motion n , then
