THE ORBIT OP UK AX US. 19 



t for the coefficient of cos X in the preceding general term of <?, this term will 



u i 

 Decome 



Q = - k cos [/' + (? + ,' +j) ] cos [,Y + y] 

 m ' 



m k sin (jV + (i* + + J) w] sin [iY + ig\ 



i 

 If we put 



fc X*flM[/ 



*, = 2 k sin [/' 



tin- Mii 2 ln'intr extended so as to include all values of j and j' which correspond 

 to tin -,'ivcn values of i and t", we shall have for the general terms of Q a 



* \ k, cos (iY 4- <?) + k. sin (*Y + iy) 1 , 

 i ( 



nr, when w<> represent the angle HI' -\-\ij by JV, 



Tliis we are to combine with the values of and 



Z = l2pt cost*?,* 



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

 n-pivs. 'lit by [i the coefficient of t in the value of N we shall have to integrate 

 diU't -rciitlals of the form 



in \\hieh the coefficient of the time t in the angle is /* -j- *'** IjCt U8 represent by 



). the integrating factor 



Tlie formula (13) will become by these substitutions, which, though a little con> 

 |)lc\, offer no difficulty, 

 * +* 



\v +J -v +l \ 



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

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



* The indices t and/ in these equation*, are not to be confounded with the coefficients of x and 

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



