THE ORBIT OF URANUS. 11 





d'W , ^, (9" + '?-';' 



= «a" ^rv.'- + "■ 



n + 1. 



—^^02 -^ " cJa" ^^''^''+^'' (9a« + ' +"* <9a''+^ 





consequently we have for the derivatives of A from formulee (8) 



The derivatives of A being formed in this way, those of 7i are immediately 

 deduced from the equations 



dx) <9» 



When s is equal to |, A is of the form 



( r7?>"' f97/'' 1 



« (0')if + (l)'a^ + (2)'a^7|- + etc. 



I oa oa J 



The quantity within parentheses is of the same form with A, in the case of 5 = |. 

 If we represent it by A we shall have 



6A t dA^ , ,A dAl . . 



d-o \ 6'» ' / (9» ' 



e'A . . ^ dA . 6'A' 



A being the same form with A, the derivatives -- — and -^^ will be of the form 



(9), substituting | for the index i, and (0)', (1)', etc., for (0), (1), etc. 



In the case of s = 5 the derivatives are obtained in the same Avay, which is too 

 simple to need elucidation. 



"NVe have now to pass from the derivatives of h to those of , the coefficients 



of the perturbative function. The form of these derivatives will depend not on 

 whether the planet is disturbing or disturbed, but on wlicthcr it is an outer or 



