66 THE ORBIT OF NEPTUNE. 



that both the longitude and motion of the hypothetical planet are entirely 

 arbitrary. 



For the differential coefficients of the elements with respect to the heliocentric 

 co-ordinates, we have 



$^ = 1 + 2 Z; cos Z + 2 7i sin I. 

 de 



dv dv 



dn ' de' 



~Tr^ — 2 cos ? — # 7i sin 2 Z — | ^ cos 2 ?. 



dli ■' ^ 



^ = 2 sin Z + I Z; sin 2 1 — -^7i cos 2 I 

 dk ^ ^ 



- -Y-^^fc sin I — h cos I. 

 a de 



1 fZr _ 2r t dr 



a dn a an a da 



1 dr 



- ~n~^^ — sin ? + 7t — Z; sin 2 Z 4- 7i cos 2 I. 

 a dn ' 



1 dr 



- ^T- = — cos Z + 7^- — 7t sin 21 — k cos 2 I. 

 a dk 



In accordance with what has been proposed, we shall substitute for s and n the 

 quantities x and a^, connected with them by the relations 



X ■=! £ -\- ah -\- pk 



sc! = n-\- a'Ji + (3'k (^ 



a and /3 being approximately the average values of — 2 cos I and + 2 sin I during 

 the last nineteen years, and a' and /3' the average values of 2 n sin I and 2 n cos I 

 during the same time. We shall take 



a = — 1.77 a' = — 0.018 



/? = - 0.85 (3' -+0.073. ^"^^ 



Then, considering v as a function of x, y, h, and k, and enclosing the new dif- 

 ferential coefficients in parentheses, we have, by suitable transformations, 



(dv \ dv /dv \ dv /dr \ dr /dr \ dr 

 dx ) de ' \dx'/ dn ' \dx / ds ' \dx'/ dn 

 (dv \ dv dv 



\dh)^cm-^''^'''^d^ 



/dv\ dv ,„ „ , dv ,n\ 



Kdkhdk-^^^^'^dT (^) 



dli ^ a ds ^ an 



1 /dr\ _ 1 

 a \dh / a 



