16 THE ORBIT OF URANUS. 



We have, therefore, for all values of i different from zero 



^'=^-'= (i^lJlW^ {^-t -WJP^ + S 2H.+l)(42)(^+3) - ^''- } (^^^ 



To obtain the value of ^^o we remark that the only constant term in cos u arises 

 from the term — es\n"g; its value is therefore — |e. The constant term in 

 ^ =; cos u — e is therefore | e, whence 



p, = _3e. (15) 



The values of q^ may be obtained in a similar way by developing sin u by 

 La Grange's theorem. But the development is rather more complex, and it is 

 easier to derive them from p^. Let us take up the equations 



= cos u — e 



yj = sin ?t 



u — e sin u = g 



Considering u, like ^ and >;, as a function of the independent variables e and (/, we 

 have by differentiation 



du d (e sin u) 



Comparing («) and (JS) 



(a) 



Q>) 



Putting in this equation for ^ and yj their developed values this equation becomes 



- . . . ~ d (eg,) . . 

 Z ipi sin ig =2 — ^-^^ sm ig 



de 



which gives by equating the coefficients of sin ig 



I 

 ?' = . 



Spide. (16) 



