GENERAL INTEGRALS OF PLANETARY MOTION. H 



(a„ a) =X (^-^ ^fi _ ^ ^) (14) 



we shall have 



0'., «0 '"^ + (a. «.) ^!^ + etc. = ^ _ ^i' g^ ^II, (15) 



dt ^ at da^ i=i 6r 6a^ 



the sign 2' including, as before, not only all values of i from 1 to n, but the cor- 

 responding terms in y; and ^. 



By giving Jc all values in succession from 1 to 6n, we shall have a system of 6^ 

 differential equations, the integration of which will give the values of the 6n 

 quantities 



in terms of the time. 



By the fundamental assumption with which we set out, the expressions for ^, 

 y;, and ^ are such that the right hand members of these equations are small quanti- 

 ties of which we neglect the powers and products. We may, therefore, after solv- 

 ing these equations so as to get the derivatives in the form 



dtti J , ,. 



-— ' = y ('/i, % a„„ t), 



dt 



integrate by a simple quadrature, supposing a^, an,, etc., in the second members to 

 be constant. Moreover we shall require the values of the quantities (%, o^) only 

 to the first degree of approximation, and within this limit they must necessarily 

 conform to the well-known law of Lagrange of being functions of the constants 

 only, and not containing the time explicitly. This theorem will materially assist 

 us in their formation. 



§ 4. Formation of the Lagrangian Coefficients («;, a^, and Reduction of the 

 Equations to a Canonical Form. 



Restoring the two classes of constants represented by a and Z, we shall have three 

 classes of the functions sought, included in the forms 



(«i, ttj), (4, Ij) and {a„, Ij). 

 Let us now differentiate the equations (12) with respect to the time, putting for 

 brevity 



ifii + 4&2 -|- -[- i^,fi^,^ = h 



*l^l + h^~2 + + is,?^zn — N 



jA -i-jAi- +jlAn = y 



we shall then have, omitting the index i of Z*, A-, and N , 



c'. = — Shh sin N 



Yl^= Shh cos N (15') 



C^\ =: Sh'lc COS N'. . 



To form the combination (%, o^) we must differentiate the equations (12) and (15') 

 with respect to «, and «,„ and substitute the results in (14). In forming these 

 quantities, two series of terms represented by the sign S of summation are to be 



