THE ORBIT OF URANUS. 



19 



?«■ 



Ic for the coefficient of cos N in the preceding general term of (?„, this term will 



become 



m 



Q,= — A; cos [j'J + (i' + i +y) 0)] cos [17 + Ig] 



a 



m 



k sin [fj + (i' + i -|-y) cj] sin [/V + i(j] 



If we put 



h = y7c cos [y'co' + (/' + i -\-j) ,>l 

 k, = 2 k sin U'cS + (*•' + i +y ) (.], 



the sign 2 being extended so as to include all values of j and j' which correspond 

 to the given values of t and i\ we sliall have for the general terms of Q^ 



"- j /.-, cos {17 -f ifj) + k, sin (i't + i<j) i , 



or, when we represent the angle i'l' -\- ig by N^ 



Q, = "- I l\ cos Ny 4- I; sin NA . 



This we are to combine with the values of ^ and ■/7 



^=:|2i>,cosiV,* 

 >7 = 12*7, sin y<7, 



in the general integral formula (13). If we substitute them m this formula, and 

 represent by i^i the coefficient of t in the value of N we shall have to integrate 

 differentials of the form 



COS \ / 



in which the coefficient of the time t in the angle is n -\- in. Let us represent by 



Vi the integrating factor 



n 



[I -\- in 



The formula (13) will become by these substitutions, which, though a little com- 

 plex, offer no difficulty, 



^P 



1 



TO ar 



+ CC 



2L- I'il.' X 



\ v^,-v^, \ \ /.-, COS [^r +(;4-y)j/]+z-,, sin [AT_+(/+y )^] \ 



-{-\v+,-v_j\ 1 A-,cos [N,^(^i-j)(j]-^k,sm [N,-\-(^i-J)g] J 

 + J v+j-v^i \ \ h cos [A^-(i-y ).9]+A',sin [N,-{l-j)g] \ 

 + \v_,-v_,\ \hcos [^,_(,-+y)^]+A-,sin lN,-(!-^j)g-]\ 



The sign 2 of finite integration here includes the separate combination of every 

 value of i with every value of /, except those combinations which make the 



* The indices i and./, in tliese equal ions, are not to be confounded with the cocfBcients of x and u 

 in the general terms of B and Q. We need not use the latter at present. 



