20 



THE ORBIT OF URANUS. 



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

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



To find, from the expression, the coefficient of the sine or cosine of cosine N L -{- ug 

 in r*$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 u ; 

 " third, i -\-j = u . . j = u -\- i ; 

 " fourth, i j = u . -. j u i. 



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

 from oc to -f-oc, i and u will also be independent, and include the same range 

 of values. Substituting for j its value in u the coefficient of 



1 m'a r7 " 



becomes 



16a L (l -f TO) 



[k. cos (JV; -f ug} -f Tc, sin 



Pi 

 + Pi ?<*-> ("i 



"<-<>) 



this expression reduces immediately to 

 l^w >(" "< o) 



Since = 



or, substituting i u for i in the second line 



22 



Hence, writing N instead of 



. __ _ c (19) 



o Ctj( 1. I 771) 



This expression fails for the particular case N= ug, where the value of v_ u will 

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



m 

 a, 



cos ug -\- A sin ug), 



and substitute in the general expression (13) it will be found that the terms in rffy 

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



, t i m'ant < 1M ^~ 1Mi ,-OAN 



r l oo =. 5 p?*.P*i C*2" I \^v) 



The two parts of r, 2 5p thus found include all the terms of the first order with 

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

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

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

 pf these terms as depend on any angle JVby 



= (k c cos N+ Tc, siu N) 





