becomes 



20 THEORBITOFURANUS. 



coefficient of the time under the sign sin or cos vanish, and so render the corre- 

 sponding value of v infinite. Tiiese cases have to be treated separately. 



To find, from the expression, the coefficient of the sine or cosine of cosine Ni-\-itg 

 in ?•," ^p, we put, in the four lines of this equation, as follows : 



In the first, i-\-j=u .•. j= u — i; 



" second, i — j^^u .-.j^ i — m; 

 " third, — i-\-j:^u .-.j^ tt-{-i; 

 " fourth, — i — J = w . • . y = — u — i. 

 In the above expressions i and j being independent, and including all values 

 from — oc to -|-oc, i and ?t will also be independent, and include the same range 

 of values. Substituting forj its value in a the coefficient of 



1 J^^^ i;;,^ cos (iVi + ng) + k, sin {N, + ug)} 

 Pi ?(»-!•) (''(«-()— '>'i ) 



1 + i'i'ZoM-i) (»'(»+«)— ^-« ) 



• [ + p. ?-(«+.■)(''-.• — "(«+>))• 



Since qj ■= — q_j this expression reduces immediately to 



or, substituting i — u for i in the second line 



22 {p.qa-n) + l\i-u)qd O'i — Vu-i)- 

 Hence, writing iV instead of Ni, 



1 ' —2 



^P=Q Ti^!' . '^rU2)^q^t-u)+Pii~u) qd (''(—'•(«-,)) lhcos{N-{-ug)-\-Jc,sm{N-^ug)] (19) 

 ' oai(l-t-m) 



This expression fails for the particular case N= ug, where the value of j'_„ will 

 be infinite. If we take each term of Q of the form 



— (A-^"* cos rtg -\- k'"' sin iig), 

 a, 



and substitute in the general expression (13) it will be found that the terms in ?'i'^p 

 which have the infinite values of v as a factor are to be omitted, and replaced by 



r,^8p = 1 ~~ 1 '7 2i>,. t:^-^ 2?„ Ai"' I (20) 



ai(l + OT) 



The two parts of ri^^p thus found include all the terms of the first order with 

 respect to the disturbing forces. But when terms of the second order are taken 

 into account, we shall find terms in Q proceeding from secular variation in which 

 the time appears as a factor, outside the signs sin and cos. Let us represent such 

 of these terms as depend on any angle iV^by 



Q = (k^ cos N-\- k, sin N) 



