20 GENERAL INTEGRALS OF PLANETARY MOTION. 



The possibility of developing the reciprocal of this denominator in the usual 

 way depends upon the condition that the constant term of SJc cos N is larger than 

 the sum of the coefficients of all the other terms, a condition which, so far as we 

 yet know, is fulfilled by all the planets and satellites of our system. Representing 

 this constant term by A-q, and the quotient of the sum of all the other terms 

 divided by k^ by A, so that 



;S';fccosi^:=^o(l + A) 



the developed expression for li will be 



n = 2^^(l-lA + ^A^-etc.). 

 When we develop the powers of A this equation will reduce itself to the form 



n = Sh cos (i\Xi + inl.^ + %^S + + hn^Sn), (33) 



each /I being, as before, of the form 

 while in each term 



«1 + *2 + *3 + + hn — 0. 



To form the second part of £lj and of £t!j in (23) and (24) we have to differen- 

 tiate the expressions (12) twice with respect to the time, and once with respect to 

 the arbitrary constants which enter into them. Putting, as before, for brevity, 



b — ifii + 463 + + Hnhn-, 



we have 



(34) 





— SPTci cos N 





— Sb% sin N 





— SV^Tili sin N'. 



For the other derivatives which enter into D!j we have 



^' = — Sij hi sin N 



||'= Sijh cos N (34)' 



— = Sjjh't cos N'. 



Forming the sum of the products which enter into Hj, in the manner represented 

 in § 4, it becomes 



'i" 4. S, \ ( ijkd V {h%i^ sin (N,—N^)- 

 » = i 1 



+ hUj^'il i^'f^d^ (sin (N'. - N'^) - sin iN\ + N'^)) | . (35) 



