With these substitutions, we have for the values of T, Y, W, the following expressions : m e ?/i(M + m) , / 1 1 . T, if _i-^ m ^ r 1 = i -}- )ft)/l I p ^7 I / COb LUb o I 5 / - r 2 p 3 r*J p A // --IV cos& sin^-O- Vp r 3 } - ( ----^V sin V. It will be noticed that the first term of the expression for T con sists of the force which corresponds to the purely elliptic motion. The two remaining terms in T, as well as the whole of V and AV, depend upon the disturbing force, as is evident from the circum stance that they contain in 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 b = l and sin b = b . 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 dis turbing 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, be the polar co-ordinates of a celestial body which moves nearly iniformly in a nearly circular orbit. The form of the orbit may be expressed by an equation of the type /( P ,0) = 0. 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 and by elimination of t from these equations the ordinary equation in polar coordinates is ascertained. With reference to the forms of the functions / x and / 2 , we shall make an assumption. Letxi, Xa> &-> be arbitrary angles ; 1( ca, 2 , &c., arbitrary angu lar velocities ; a,, a 2 , &c., small arbitrary linear magnitudes ; and ?; 2 , r. 2 , &c., small numerical factors, a is an arbitrary linear magni tude, and w is an arbitrary angular velocity. We shall assume that p=^a + a L cos (<V + Xi)+ a 2 cos (^ + Xa) + > & c - = cat + TJ ! sin (u^ + Xj) + 1? 2 sin (ta.J, + Xz) +> & 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 p 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 con dition is fulfilled in the form we have assumed for p. Under all circumstances / = !!>TiToT, &c. ; p = <jia l a. 2 , &c. ; and as a v a. 2 , &c., are small quantities it appears that p is neces sarily restricted to narrow limits. By similar reasoning we can justify the equation for 6, for we can show that the angular velocity of the celestial body to which it refers must be approximately uniform. By differentiation, de = o> + T^W! cos (oi-ft + Xi) + i?j*-2 COS Under all circumstances we must have ^ = > T T dt ^o+W^W: = <fco w e& " 1&>1 2 : and as rj^, ri. 2 , &c., are all small quantities, it is obvious that - i s ctt 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 0, it is to be remarked that the number of dis posable constants a 1} a. 2 , &c., xi> Xa> & c -> ^u ^a; ^ c -> w i> u v ^- c> a, to 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 a + 2a 1 cos ( = o>t + 2,ri 1 sin 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 pro ducts of the small quantities a x , a. 2> &c., TJJ, 7j 2 , &c., may be dis carded. T is to be computed from the well-known formula where we have dt de__ dt dff 2 df 1 de* whence by substitution T - maa? - 2i( 1 c> 1 2 + 2Tj I rtcoo> 1 + a^) cos (co 1 < + Xi)- To compute V, the force perpendicular to the radius vector, we proceed as follows : p * = a? + 22<mj cos dO = ~dt whence we find %dO = 2 P dt " Differentiating we have Substituting this in the ordinary formula v _? d ^ ~ we have finally V= - 2w(2a j coco 1 -l-; 1 ffltf 1 2 ) sin Let us assume, for the sake of brevity, A! = aj + xi ; A 2 = o>. 2 t + Xa 5 &c - &c. &c. then the result to which we have been conducted maybe thus stated: If a body be moving in a nearly circular orbit under the infiv.sace of a radial force equal to - mrtoi 2 - i2F 1 cos A] and a force perpendicular to the radius vector equal to sin then the path which the body describes is defined by the equations r = a + 2 L cos X : ;
§ 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 1} a. 2 , &c., 77 1( rt. 2 , &c., we have r = a; = wt; T=-maca"; 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 uni-
