41] ON THE GENERAL VALUES OF THE OBLIQUITY OF THE ECLIPTIC. 305 

 or, substituting from above for cos &>, sin <o cos \ and sin w sin X, 



-^ = - c ~ {cos + sin tan & cos (< - <')} {sin - cos tan 0' cos (<f> - <')}, 



dd 



-j- = c cos 2 6' {cos + sin 6 tan 0' cos (< 0')} tan & sin (< <'), 



which are the differential equations for determining and <f>, 6' and <' 

 being supposed to be already known in terms of t. 



From the above we may deduce the following : 



d /cos a)\ da , . n , \ dp , . 



dt (cos yj = ft (sm * C S *> + ft (sm * Sm *> 



The integration of these equations may be readily effected by the method 

 of indeterminate coefficients. 



Suppose the values of p and q to be 



2? = Sy< sin (#/ + &), 



where takes the successive integral values 0, 1, 2, &c., equal in number 

 to the number of planets considered, and the quantities y f , g i} and /8 t - are 

 known constants. 



Then we may find that 



e = h + % tan 7tSi (% - 1 ) y/ + 1- cot AS (a,. - ^) y/ 

 /y t cos {(k -g^t + a- ft,} 



i cos 2 {(& -g t )t + a- fr} 



^00 cos 



+ SaVo'; cos {(</, -/)< + A - A')- 

 And 



<f> = kt + a + Ib^ sin {(k -g^t + a- A} 



+ S6 (y,) 2 sin 2 {(i -g t )t + a- fr] 



M sin {(2k - g t - 9j ) t + (2a - A - A)} 



'^oo- sin ((9i - ft) < + A- - A)> 



in which i and ji are supposed to be different integers. 



39 



