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 



(0)a'-^6<P + (l)a*+'^ + (2)a'+^^£^+etc. + a— ^-^, (8) 



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

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



V'-') 



and h'l^ being, as usual, the coefficient of cos i^ in the development of 



(1 — 2a cos ^4- a^)-" 



in multiples of cos 4), 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 ¥*J had been taken with respect to » instead of a. In fact the 



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



the form 



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

 be of the form 



a- '' ( (Oy WJ + (1)' ^ + (2)' -^ + etc. ), 



dA d'^'A 

 from which the derivatives , ^, etc., may be found with great facility. 



As in the actual developments of R wliich 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 formulae 



D\ = a{Da -\-aD\). 

 Beginning with the case of s = J, we have 



dv da 



'W~'' da '^"' da' 



