X THE THEORY OP THE LONG INEQUALITY 



the eccentricity or some particular combination of eccentricity and inclination. And let 

 p cos (k'n't + q) be a particular term in a;' : then - p'k' 2 n' cos (k'n't + q') will be the cor- 



responding term in — — , 



dV 



.'. the coefficient of their product in ®~t^ = — pp'k' 2 n' 2 , 



, d~~v , 



and the coefficient of the product in a -— = — pp k 2 n 2 , 



OjZ" 



k' 2 n' 2 

 the ratio of these coefficients = — — -, 



Jen 



and the same ratio holds for the similar combination in « — - and «' , and also for anv 



* df * df ' 



... . d'z' , , <Fz 

 combination in » — - and % — . 

 df df 



sin sin 



Hence the coefficient of the product (knt + q) (k'n't + a') in R bears to the 



■ cos cos 



corresponding coefficient in R' the ratio 



tn' l + m fc'V 2 , . , , m'k' 2 a? 



— ; .. . which equals — , „ >Q 



m l + m' k*n 2 ^ mVa 3 



accurately : and this is the ratio of the coefficients of the terms involving (kn ± k'ri) t to 



,-ii ... sin sin , , , 



which the combination (knt + q) (knt + o ) gives rise. 



cos cos 



Hence, if m'M and mM' be the corresponding coefficients in the unsymmetrical parts of R 



and R', 



M' k* 



k i 

 or in the case of the long inequality, in which — = - , 



fC A, 



M' _ l 

 ~M ~ te 5 ' 

 (Since this ratio is nearly = 1, it follows that the usual approximate relation between the 



■ ■ i ,.11 • i- i • i C »«' , 



principal parts of the long inequality m mean longitude, & m j or, more accurately, 



r— holds true when some of the terms arise from the unsymmetrical part of R, but the 



mk a 



ratio is not sufficiently exact to be of much use except in small terms depending on the square 

 of the disturbing force.) 



10. Making use of the relations given in the last article, the coefficients for Neptune may 

 be calculated from those of Uranus by the following formulae, 



aM t , , aM 2 



, _. aM x , ... aM 2 ( 1 \ 



a a \ ia J 



, , aM* , , aM 4 aM b 



aM 3 = a Mi = a Mr, = , 



a a a 



