22 GENERAL INTEGRALS OP PLANETARY MOTION. 



and before the second one 



ji=cc j-2=cc i3,.=oc 



2 2 2 



J, =— oc i2 = — oc >. = — oc 



Differentiating (33) with respect to Ij, we have 



^ = — Sijh sin N. (36) 



By the substitution of these expressions (23) now assumes the form 



^= — Sh'j sin N, (37) 



putting for brevity 



h' = ijJi -\- Hj. 



By the fundamental hypothesis that the adopted expressions for ^, 57, and c, are 

 first approximations to the true values of those quantities, it follows that in adding 

 (35) and (36) all the terms which are not of the order of those neglected in the 

 first approximation destroy each other, so that K is of the order of the quantities 

 neglected in that approximation. 



To form the equations (24) we diff'erentiate (12) with respect to c, whereby, omit- 

 ting the index i with which ^, >/, ^, k, and Ic are always to be considered as affected, 

 we find 



^= ^^cosi\^+<Mf sin^ 

 CCj oCj dCj 



^^=sfsmN^t8k^^cosN (37)' 



CGj dCj dcj 



6^ „ dk' . ,-,,,, r,^ Sb' ,7-, 



TT" = 8 ^ sm N' 4- t S'Jc -;- cos N'. 

 The sum of the products of these expressions by (34) which enter into (24) is 

 — ^ i" S\, I (hlc\ fl cos (N, - N,) — t ^bVc)^ p sin (N, - N^) 



+ K^'^^')^ 1^' (cos {N', - N'^) - cos (N\, + iV;) ) 



- h t {V'^X ~ (sin iN,—N;) - sin {N\ + iV;) j , 



while by differentiating (33) we find 



^^=S(f- COS N-th^^.mN). (37)" 



Taking the difference of these two expressions, the equations (24) will assume the 

 form 



^^^i =—S7i" cos N^t Sli" sin K (38) 



the quantities 7i" and If being formed by a process similar to that used in forming 

 K. We have now to integrate the expressions (37) and (38), and substitute the 



