12 THE ORBIT OF NEPTUNE. 



Substituting these expressions for the differential coefficients in the values of 



. 

 ^ and rfr we have 



- =. h sin y> sin N { (i' +/') cot y + (i -\-j) cosec y } - -j- cos \y cos it cos N. 



(id) tt Q> du 



- 

 (id) 



1 dR 



m 

 

 (t 



sin <> dQ 



Let us now put 



.. , ,. 

 i' +/)co 



m dli 

 cosecvj i ~, - r - 



It may be remarked that i will then be the coefficient of the longitude of the 

 common node of the orbits in the usual development of the perturbative function. 

 The above equations may then be put into the form 



dR m -I , T m , 1 . f. I 1 I AT 1 m 'fa I 7 



-r = rih cosec y sm / sin N -7 (V + f) ft tan A y sm x' sin N 1 ? -5- cos -1 y cos x cos N. 

 dip a! a ^ a du 



1 dR m , . . , r wi , ,, , . . ,,. m dh 



-~j- th cosec y cos x' sin IV -(- (t T +/) A tan J, y cos x' sin /v y cos 



sin <f dtf 



Substituting these expressions in (6), and integrating, we shall have the values 

 of && and &$', the perturbations of the inclination and node. 



For the perturbations of the latitude, counted in the direction perpendicular to 

 the plane of the orbit, we shall have 



Where 



Putting 



$(? = %$ sin (^ 0') sin $W cos (tf 9) 

 = mv sec ty{T+I} sin (N+ V) 

 + mv sec 4/ { T 1} sin f^ F) 



T \ ~ cos } y ; /= ^ 7i { t cosec y + (t v +/) tan J y } 

 V= true distance of planet from common node. 



i T 



i * 



(7) 



and developing Fin terms of ^, and/ to terms of the second order with respect 

 to the eccentricity, we shall have 



+ <f sin (#;+*,+ /) 

 d sin (N'+ % /) 



-f- 



sn 



T , /) 



(8) 



