36 



THE ORBIT OF URANUS. 



on the sun. They are each of the form N X a~ 2 , N being a numerical coefficient 

 given by Le Verrier under the coefficient for each term. The derivative of this 

 expression with respect to is 2N X a~ 2 , so that for the corresponding terms 

 in DJi and Dili, we have 



ADJi = 2.1/t 



The values of h and its derivatives, corresponding to any one argument $ and i, 

 are to be combined into two terms depending the one on the cosine, the other on 

 the sine of the argument. Let us represent by g the mean anomaly of Uranus, 

 and let us put I' for the mean longitude of Saturn counted from the perihelion of 

 Uranus, or, more exactly, for the arc X a. Put also 



P"=j\ 



Then, for each value of N there will be several values of P corresponding to dif- 

 ferent powers and products of the eccentricities and inclinations in h. Distin- 

 guishing these values and the corresponding values of h by subscript numerals, we 

 shall have a series of terms of R of the following form 



7i, cos 



m 



<*\ 



+ etc. 



etc. 



and by putting 



li e = 7*! cos P\ -\- 7*2 cos P' 2 -f- 7t 3 cos P 3 -(- etc. 

 '/, = h t sin P\ 7 2 sin P'.j 7^ sin P' 3 etc. 



(41) 



The above terms may be condensed into 



m 



m 



. 



H = h c cos N-\ -- n. sin jv, 

 a t ! 



which are of the form supposed in the preceding theory. 



In order that the derivative of R, with respect to the true longitude of Uranus, 

 may be expressed in the form 



8R m m 



---- = v, sin N -\ --- V..COSN 

 <9v j o, 



we must, by (7), put 



v, = (i -j-ji) A! cos P\ (i -}-fa) h-i cos P\ etc. 

 *>c = (i -\-J\~) A, sin P\ (i -f / 2 ) 7* 2 sin P' 2 etc. 



(42) 



y,, y a , representing the several values of j in the different terms which correspond 

 to one and the same set of values of t and f. 



