12 THE ORBIT OF URANUS. 



inner one. Let us then suppose for the present, that a and » refer to the inner 

 planet, and put r>^ for the logarithm of the mean distance of the outer one. We 

 then have for the derivatives relatively to » 



a, 1 d^h 



din" Oi ^»» ' 



and for the first derivative relatively to », using the symbolic notation. 



6^ 



^ = ~{D..-l)h. 



The symbols in the second member being distributive, we have by successive 

 differentiation 



-^ = l(i)., -1)"^. 



The quantity h is a function of a, the ratio of the mean distances or of C' — 'i, G 

 being the neperian base. Hence 



Dvih = — Dvh, 

 which substituted in the last equation gives 



This formula gives for the first two derivatives 



o 





d 

 h 



6' 



«i _ J_ (f/, I 2_£^ _|_ ^'^ \ 



Substituting in the general formulae (7) these expressions for the derivatives 

 relatively to » and -d^ we have expressions for the derivatives of K relatively to 

 V, V, p, p', it being understood, however, that all the quantities are expressed in 

 functions of the elements of elliptic motion. 



In order to compute the perturbations of the second order we must carry R and 

 such of its derivatives as enter into the differential equations (1) to quantities of 

 the first order with respect to the perturbations. Let us then represent by v,,, Vq', 

 Poi p'oi 7oi the elliptic values of v, V, p, p', and y, which we have assumed in the 

 iirst approximation to the perturbations, and by 5v, 5v', etc., the quantities to be 



