24 GENERAL INTEGRALS OP PLANETARY MOTION. 



We have next, in the second of equations (40) to substitute the expressions for 

 the derivatives in (37)' and (37)", retaining only the terms multiphed by t. This 

 gives by substituting for h its developed expression 



6 = 'i'l&i + 4^2 + + %„ 63^ 



+ 'S [^■^H^^ ^ + H ^+ + *-^4")| cosiNT 



+ ^ I 2^ J.{j, y^+J.^+ +^3.^^) I cos N. 



Adding this expression to (41), we find that the sum reduces to a series of terms 

 each of vs'hich has a factor of the form 



dCj 6ci ' 



By (31) these factors are all zero. Hence the terms of (39) multiplied by i destroy 

 each other, and we have 



the parenthesis around --^ indicating that all the terms multiplied by the time in 



that expression are to be omitted ; in other words, that, in taking the derivatives of 

 ^5 ^5 >?! and ^ with respect to Cj, we are only to consider the coefiicients 7i, Jc, and 

 k' as functions of these quantities, and are not to vary b^, ho,, etc. 



§ 7. Form of the Second Approxiynation. 



The rest of our process is now as follows : By integrating (37) and (38), the 

 last member of (38) being omitted, we have 



hcj = 8^'^^ co^ N 



{hlj) = — 'S^ sin N. 



The co-ordinates ^, 57, and c, in (12) being expressed as functions of the quanti- 

 ties Cj and Jp we are to suppose these quantities increased by their perturbations, 

 that is, we are to find 



or, since we have replaced ?; by "Ki, 





