10 GENERAL INTEGRALS OF PLANETARY MOTION. 



^i = Ski COS (i,l^ + ip.^ -j- + hn^sn) 



Yii = Ski sin (i,;ii + 'i2'^2 + + *3,A3«.) (12) 



^i = Sk'i sin (/A +^;:i2 + + J3,AJ, 



in which all the quantities are supposed to be given in terms of 6n arbitrary con- 

 stants and the time, each X being of the form 



Ai = /j + hit, 



?i being an arbitrary constant, which each b, h; and k' is given as a function of 3?j 

 other arbitrary constants, which we may represent in the most general way by 



So long as no distinction between a and I is necessary, we may represent the 

 entire 6u arbitrary constants by 



Let us now take the complete second derivatives of (12) with respect to the time, 

 supposing all 6n constants variable. We shall suppose the variable constants to 

 fulfil Lagranere's conditions, now 3w in number : — 



which will give 



^_^_C'. etc 



From the second derivatives, combined with the differential equations (11), we 

 shall have 3?i equations of the form 



j^i daj dt "^ e^i ~6f' 



which it is required to satisfy. The expression in the right-hand member of this 



dR 



equation corresponds to -^ in the usual theory, when R is the perturbative function. 



Let us multiply this equation by -j^, and add up the 2)n equations which we may 



form in this way by substituting for ^^ all the values of ^, r,, and f in succession. 



We may thus obtain 



,„„ ,.^0. 6^dj^c^_d£l_ '='f d% d^ 

 i = i j = i da,, dttj dt da,, i = i dt^ da,,'' 



the sign 2' indicating that all values of 97 and ^ as Avell as of ,? are to be included. 



dR 



The right-hand member of this equation corresponds to v" in the usual theory. 



Let us now multiply the equations (13), the first by ^, the second by ^', and the 

 third by Jti , and add together the Sn equations which may be thus formed by giving 

 i all its values. If we subtract their sum from the last equation, putting 



