TIM: OIIBIT OF URANUS. 57 



ing formula 1 . Kxpress the usual <li \ ( lopmrnts of the longitude and logarithm of 

 radius vector in the form 



v = I -f- 2 V t sin ij ; 



f = -j- 2 RI cos iij. 

 Put also 



v.. ''". 



Kxpress any set of corresponding terms of the preceding perturbations in the form 



1 tin = F. sin JV-- F, cos N; 

 le = E e cos N+E. sin N; 

 5 = A, cos W-f- ji, sin N. 

 \\ -hall then have 



= 2 ( F' ( .F, + F, E c ) sin (N + tV/) + 2(^1^,- V, E e } sin (A^- 1^/) 

 + 2 ( V, F e - V, .) cos (JV^+ 1 -f 2 ( F', F e + F, .) cos (N- ig) 



- 7T, F.) cos (AT+ t^) + 2 (R' t E e + 7T, F, 

 2 (7?, . + 7T, 7-' e ) sin (N+ iy) + 2 (R t E. - IT { FJ nu(N- iy) 



The numerical values of F, F, 7?, and 72" arc as follows : 



F, = 



F,= 



F s = 



F 4 = 



7? = 



R l= 0.99753 JZ"i = + 0.9991 7 



R t = 0.07020 JS" a = + 0.07030 



R t = 0.00466 72*, = -j- 0.00467 



The final results of the entire computations are given in the following table: 

 In the columns 5t?,, we have the complete perturbations of the longitude computed 



r)K f)R 



by the direct method from the values of ^, - -, Q, etc., already given. Next 



<?v op 



we have, under the caption tr 2 , the perturbations of the true longitude deduced 

 from the long period perturbations of the elements, as set forth in the last para- 

 graph, omitting the constants added to the perturbations. Under bv t we have the 



8 April, 1873. 



