82 LECTURES ON THE LUNAR THEORY. [LECT. 



Then our equation may be integrated 



dxd8x = 8x+ 



dt dt dt dt dx dy 



where T = 



so that T is a known function of t, which involves an arbitrary constant. 



Now let us assume 



8x v cos (f> w sin <f>, 



8 = v sin < + w cos <). 



dx dy dSx dy c , 



Substitute above for - , , , ; we find 



= - 



dt) \dx dy 



dR dR . , dV 



But --T- cos (f> + T sin < = 7 , 



oa; at/ a< 



dR ,,/d$ , A 



j-sinc&H--r-cos9= F -77 + 2%' . 

 CMC ay \a< / 



Therefore 7-u, -*+ F + 2n' , + T, 



cw \ai / 



T7 , , T 



or F -r - -r.v= 2wV(~g+ ri\ + T, 



at at \dt / 



v [2 fd<f> ,\ (T , 



whence y= I -p- ( 7. + n 1 >CK + I -y., dt. 



An arbitrary constant is included on the right. This equation shews 

 that when w is known, v can be found ; it remains to determine w. 



Now by actual differentiation 



. d8x , dSy dv d<h 



. d*Sx . d?Sy d*iv dv d<f> /dd>\ 3 



-sin (-775- + cos 6j =-j^ + ^ -r-^-iv(~} +v 3%. 

 dt* df df dt dt dt 



