XXXV111 



THE THEORY OF THE LONG INEQUALITY 



= - 2"-673 sin 2X + 0"-071 cos 2\. 



a m " 



.-. 9gm + l"-950 sin 2\ - 0"'052 cos2X.* 



46. The second term in the expression for It in Art. (39), is of the order — at least ; 

 we may therefore put R = J2, = m (P cos X - P 1 sin X) ; 



+ | [— J K^ 2 - P' 2 ) cos 2X - 2PP' sin 2Xf ; 



'i*fi„i 



3mVo 



2co 3 sin 



— , {(P 2 - P' 2 ) sin 2X + 2PP cos 2X} ; 



therefore neglecting the secular part, 



ttgm - o"-137 sin 2X + 0"-042 cos 2X, 



4m' 



^r = +^7i al ^ nearl y' 



m 



.-. 3^- - 0"'143 sin 2X + 0"'044. cos 2X. 



47. $R will also contain terms involving X, which arise from terms in R involving X + It, 

 combined with terms in the variations of the elements involving It, and vice versa ; but these 

 terms are not of much importance. We shall only calculate the following, which arises from 

 the double combination of the arguments 



2 (nt + e) - 3 (n't + e), and (nt + e) - (n't + e). 

 Let <j) = 2 (nt + e) - 3 (n't + e) + •ar, 

 <p! = 2 (nt + e) - 3 (n't + e) + -ar', 

 and R = m'Z^e cos <£ + m'D 2 e cos d/, 



where aZ) 



1 = i( 



6bW 



+ a 



dbji) 

 da 



dbS* 



)• 



also let /? y = m'C cos \|^, where yjs = nt + e - (n't + e'), 



• If the expansion of <5J? be carried a step further, we shall 



have the terms of the third order ' <5ie4 1 «r+2-r-r( s i'Er) i! + 



dedur dor 



similar terms depending on £,e' and a,W. Retaining only the 



principal part of the former, we shall have 



9m''n 3 a 3 IdP* dP'*\ 



K- 



ieui \ de % 



dP'' \ 

 de'J 



ftfteSS 



= + 3" .89 sin X - 0"-81 cos X, 



2m 



and f? = <x». it = - 2". 84 sin X, 



m 



rejecting the second term which is less than 1". The terms 

 depending on S^, £i«" are less than 1", and may therefore be 

 rejected. 



