THE ORBIT OF NEPTUNE. 11 



The values of the second of each pair of differential coefficients can easily be 

 determined geometrically. /I, w, /I', etc., it will be remembered, represent the dis- 

 tance of certain points on each orbit from the ascending node of the disturbing 

 planet on the disturbed one : the infinitesimal changes in those quantities, produced 

 by infinitesimal changes in the position of the plane of either orbit, will be due 

 entirely to the changes in the position of that node. Let us put 



x 1 = distance of common node from ascending node of disturbed planet on the 

 ecliptic. 



x zz same quantity for disturbing planet. 



By drawing the diagram, it will readily be seen that by a change in <>' the 

 common node will be moved forward on the disturbed planet by the amount 



+ sin x 1 cot yd<p', 

 and on the disturbing planet by the amount 



+ sin x' cosec yd<p', 

 while Y will be varied by the amount 



cos x'd<p'. 

 In like manner, by a change in 0', the corresponding changes will be 



cos x' sin <>' cot ydQ', 



cos x" sin <>' cosec ydff, 



sin x' sin 

 We therefore have 



cZ/l da 



1 do' 



zz . -M zz cos x cot y, 



O I 1-1 ,*' fl(V I ' 



I do 



-jfT, cos x cosec y, 



dy 1 dy 



f-. = cos x'; -. , jfe = sin x'. 



' ' '' 



Also, by the differentiation of the representative term of R, 



dR mi'h . , T dR mj'li . ,, 



- 7 - - zz -- r sin N, - T - = -- -- sin N, 



d% a' drf a! 



dR mill . ,., dR mjJi . , r 



-^- = --- Bin JV, -j = -- r sin N, 



dh a' f?u a' 



dR dR du m dh 



-j- zz -, -- r~ zz \ , -T- 



dy du dy a' du 



