16 



THE ORBIT OF URANUS. 



cos u = . 



cos 



COS 



* 



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



etc. 1 (14) 



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

 from the term esin 2 ^; its value is therefore \e. The constant term in 

 = cos u e is therefore | e, whence 



jp = 3e. (15) 



The values of q t 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 



Y! = sin u 



u e sin u = g 



Considering u, like and vj, as a function of the independent variables e and g, we 

 have by differentiation 



du d (e sin w) 



de de 



= 



du 6 (evj) 



sin 



Comparing (a) and 



' de de 1 e cos u 



du_ I 



dg 1 e cos u 



du _ du <9 8u _ 



'de dg~ du dg ~ 



(a) 



Putting in this equation for and n their developed values this equation becomes 



2 ip, sin ig =2 ^^ sin ig 



de 



vhich gives by equating the coefficients of sin ig 



^it \JPide. (16) 



