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 k cos I + 2 h sin I. 

 (.IE 



dv dv 



dn ~ de ' 



dv 

 dh 



-JT 2 cos I f h sin 2 I | k cos 2 I. 



t3Q 



-TT=. 2 sin I + I k sin 2 7 | h cos 2 I. 

 ct/c 



\ dr . 



- -j /: sin I li cos t. 



ae 



1 d> _ J^ t_ dr_ 

 a dn ~ 3 an a de 



7r sin I + h k sin 2 I + A cos 2 I. 

 a dh 



z= cos 7 + Z; 7i sin 2 Z k cos 2 Z. 

 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 =. e + ah -\- (3k ... . 



af = n + a'h 



a and ft being approximately the average values of 2 cos Z and + 2 sin Z during 

 the last nineteen years, and a' and ft' the average values of 2 n sin Z and 2 n cos Z 

 during the same time. We shall take 



a = 1.77 a' = 0.018 



ft = 0.85 P' = + 0.073. 



Then, considering v as a function of x, y, It, and /;, and enclosing the new dif- 

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



dv\ dv {dv\_dv ( c ]^\ dr_. i 1 ^] 

 di ' Vcfcey dn ' \dx / de ' \dstf / dn 



dv 



/dv \ dv . dv 



( -71- ) 77 ( + -j- 



\dh / dh ' de 



