26 GENERAL INTEGRALS OF PLANETARY MOTION. 



it follows that all these terms will be of the same form with those already contained 

 in ^, 57, and f (12). 



In the preceding integration we have tacitly supposed the coefficient of the time, 

 b, never to vanish in any case. But some of the values of N will necessarily be 

 zero, and in this case, instead of having 



J 



Jc dt cos ]S[=^ J- sin iV, 

 we must put 



I h dt cos N = ht. 



> 



The only terms of this form are found in hi. If, in (38), we represent the coeffi- 

 cient of the vanishing term by h"^, we shall have for the terms in question 



hl= — h\t. 



This adds to 2, the same expression, and is equivalent to diminishing b by the 

 quantity h"(,. We make this change not only in the original terms of ^, yj, and ^, 

 but also in the terms of S^, St;, and 8^, because the change will only affect them 

 by quantities of the second order, which we have rejected throughout. 



Making these changes, the expressions 



^-\-^^, 57 + ^>7, find ^ + ^^', 



will now satisfy the differential equations (11) to quantities of the second order, 

 while their form will still be in all respects the same as in (12). As we have 

 made this one approximation without changing the form of the original integrals, 

 so may Ave make any number of successive approximations. We may, therefore, 

 regard the form 



^ = Sh cos (iiXi + ^2:^2 + + hn^Sn) 



Yl — 8h sin (ii^i + Slo^ + + hn^sn) , 



^ :^ Sk sin (^i^i +y2^2 + +i3,A«), 



Avhere each "k is of the form 



■ki = ?i + bf,, 



li being an arbitrary constant, and 7e, k', and b^ being each functions of Sn other 

 arbitrary constants, while 



■il + ^3 +••••. + hn = 1, 



and,/, +j;+ _j__y;, = 0, 



in each separate term under the sign S, to be a general form in which the relative 

 co-ordinates of n planets, revolving in nearly circular orbits with a nearly uniform 

 motion, may be developed when the approximations are continued indefinitely. 

 This may, therefore, be regarded as the general form of the integrals of planetary 

 nK)tion. 



§ 8. Oeneral Theorem. 



If we express the relative living force of the entire system in terms of the canonical 

 elements, the coefficients of the time bi, b^, b-j^ will each be eqwd to the negative 



