THE ORBIT OF NEPTUNE. 13 



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

 orbit, we put 



2)^sm(psm6; q =: sin <p cos d ; 



T z=: longitude of common node of the two orbits. 

 We then have 



Sj)' = 2 mv {/sin t cos N — T'cos r sin N] ; 



8q' =z 2 mv {/cos tt cos N-\- Tsin r sin N} . (9) 



Or, V= mv{{I— T) sin (N+r) — (I +T) sin (N—r)}; 



hq' = mv { (/— T) cos {N+ t) + (/+ T) cos {N—r) } ; 



§ 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 i are zero, the same expressions apply, only 

 changing 



V sin iVinto n't cos N ; 



V cos iVinto — ')i!t sin N. 



We therefore have, for the secular variations, 

 mn' L(i cosN; 



■ mn' Eq sin N; 

 mn' Wo cos N; (10) 



ell " 



eld 



dt' 



dn' 



dt' 



-^ zz — 2 m')T! \Io sin t sin N -\- % cos t cos iV} ; 



dcf 



y- -zz — 2 mil! \It^ cos TT sin N — % sin t cos N\. 



Owing to the smallness of the eccentricity of Neptune, it will be 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 j)erturbations to 

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



7i zr e sin 7t ; hzize cos n. 



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

 approximation, 



-^ z= mi}! dWo cos (iV+ 7i') ; 



k . (") 



-^ = — mil! dWo sin {N-\- n') . 



§ 9. Dei-elopment of the action of an inner on outer planet tlirough the Sun. 

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

 two distinct parts. 



