10 



THE ORBIT OF URANUS. 



The formation of the derivatives in the second member of this equation demands 

 attention. In the analytic development of the perturbative function each value of 

 h is composed of a series of terms each of the form 



EX A, 



E being a function of the eccentricities and mutual inclination, and A a function 

 of a of the form 



r)t>W i /3W 2*+2n-l -OB7(0 



(0)a'-^> + (l)a^ l ^ + (2)a'+^^+etc. + a-^^, (8) 



(0), (1), etc., being numerical coefficients connected with the coefficients F 1 *' tabu- 

 lated by Le Verrier, in Tome I of his Annales de 1'Observatoire, by the relation 



/ \ _ _ 



" O.37..M 5 



and I'? being, as usual, the coefficient of cos i<p in the development of 



in multiples of cos <>, and n 1 the sum of the exponents of the eccentricities in E. 



It would have been much more convenient if in effecting this development the 



derivatives of 6 ( * } had been taken with respect to instead of a. In fact the 



derivative - when expressed in terms of the derivatives with respect to is of 



the form 



/57>(0 (9JW cW 1 ') 



., _ ?_ I y> L_ nfp _I_ n 



' & 1 * I ltj 1 * o CwV ^^ /C 1; 



Therefore, when expressed in terms of the derivatives with respect to *>, A will 

 be of the form 



/ <% (0 <9W \ 



a-* ( (0)' jco + ( iy ^- + (2)' ^f + etc. ), 



dA d 2 A . 



from which the derivatives - , -v-g-> e tc., may be found with great facility. 



As in the actual developments of R which we possess, the values of A are given 

 in the form (8), we must find the expression for the first two derivatives of its 

 several terms with respect to , which we easily do by the application of the sym- 

 bolic formula? 



D * = aD 



Beginning with the case of s = i, we have 





da 



