84 LECTURES ON THE LUNAR THEORY. [LECT. XVIII. 



But by the equations proved at the end of Lecture XVII., the terms 

 in v cancel one another, and we are left with the equation for w: 



'3(^1 + 6n' ~ + 4ri- - 7 -^ sm' J + 2 ^^ sin <f> cos <j> - ^^ cos' 



(j I \ rri 



-,- + n' } -=r r X sin 4> + Y cos <f>. 

 dt* ]V 



Or since 



1 dx . , _ ]_ dy 



(^ 1 / rf/2 . cZ^ 



-5^ + 2n' = TT- - -,- sin 6 + T- cos 

 (/i! F\ era 



the coefficient of ry is 



3 / rf/2 CJT/ (j/? (/V _ r ^f _dJKdy dRdx\ , 

 V* \ dx dt + dy dt) ' ' V* ( dx dt + dy dt] * 



l^ \d*R (dy\* _ #R dx dy c 



V-\ dx" (dt) ' dxdy dt dt dy" \dt J 



= P, say. 



This function P is a known function of t ; it may be seen that if 



x ta i cos i (t + y), 

 y = 26 ; sin i (t + 7), 

 then P may be developed in the form 



Hence if we omit the terms X, Y, due to other disturbances not yet 

 allowed for, the equation for w assumes the form 



This is identical in form with the equation treated in Lecture XIV., 

 to find the motion of the node. The value of w may be found by the 

 method there employed, and the value of v deduced from it. 



