74 LECTURES ON THE LUNAR THEORY. [LECT. 



If we preferred to define a so that the constant term in H- were equal 

 to w 2 a 4 , we should have 



or 



3 / i i\ 



-rfci? '- m-tfa 4 ( cos 4 - + sin 4 - I , 



2, \ 2, LI 



p = na* l+-m z - 3m? sin 2 - cos 2 - . 



Let us next find the latitude and the motion of the node. 

 Suppose that *' = / + Ai, 



in which Ai, AJV are small, i is a constant, and N varies slowly in pro- 

 portion to the time, so that we may assume 



r = N, sin 2 (0 - 0'} + iV s sin 20 + N 3 sin 20' + N t sin 2(0+ 0'), 

 Ai = I, cos 2 (0 - 0') + 1, cos 20 + 1 3 cos 20' + / 4 cos 2(0 + 0'). 



Then remembering that 



dff dN 



at dt ' 



an expression that must be used in the terms of chief importance, we have 

 = -=-2(1- TO) n^ sin 2(0- 0'} - 2nL sin 20 



- 2 (m - - ~] nI 3 sin 2ff - 2 (1 + m) nl t sin 2 (0 + ff), 



\ Ib \JLJj I 



~dt = dt" + C ~dr = -1 n + 2 ( l - m ) nN i cos 2 (0 ~ 6'} + 2nN * cos 2 ^ 



\ i- ] nN. cos 20' + 2 (1 + m) nN. cos 2(0 + 0'} 



n dt / 



--sin 20' 



in which the last term will be found to be required to get the constant 

 q correctly to the order m 3 . 



rpi , i Zr cos -Zrsin0 ,. , 



Ihese must be equated to ^ . -==. r respectively. 



H Hsmi J 



