THE ORBIT OF NEPTUNE. 13 



For the perturbations of the constants which determine the position, of the 



orbit, we put 



p = sin < sin 6 ; q = sm<pcosQ; 



t longitude of common node of the two orbits. 



We then have 



$// =z 2 mv {/sin t cos N ^cos t sin N] ; 



&c( = 2 mv { I cos T cos N-}- Tsin t sin N } . (9) 



Or, f/= mv{(I T)sm(N+r) (I+T)sm(N *)}>, 



V = mv { (I T) cos (N+ T) + (7+ T) cos (Nr) } ; 



8. The equations (2) and (9) determine the periodic perturbations of the 

 elements. For the secular variations, which proceed from those terms of the 

 perturbative in which both i' and *' are zero, the same expressions apply, only 



changing 



v sin JVinto n't Cos N ; 

 v cos JVinto n't sin N. 



We therefore have, for the secular variations, 



dl' 



-jj- mri L cos N ; 



de! . 



-^ mn A) sin Jy ; 

 dt 



= mn'W cosN; (10) 



-~- =. 2 mn' {I sin t sin N-\- T cos t cos N} ; 



U-t 



^r 2 mn' { IQ cos r sin N T u sin r cos N} . 

 ctt 



Owing to the smallness of the eccentricity of Neptune, it will l>e advisable to 

 substitute the rectangular co-ordinates of the centre of its orbit for the eccentri- 

 city and longitude of perihelion. The perihelion itself is subject to changes so 

 great that it would otherwise be necessary to develop the perturbations to 

 quantities of a higher order than the first. We shall, therefore, put 



II-=.G sin 7i ; lt e cos n. 



For the secular variations of h and k, we then have, to a sufficient degree of 

 approximation, 



dk 



sn 



9. Development of the action of an inner on outer planet through the Sun. 

 The perturbations which one planet produces on another may be divided into 

 two distinct parts. 



