70 

 Therefore 



LECTURES ON THE LUNAR THEORY. [LECT. 



di _ Zr cos 

 dt~ ~H 



d-N _ Zrain 

 dt Hs'mi ' 



which give the changes of the elements required. 



Now let NMS be a spherical triangle, the centre of the sphere being 

 G, the centre of gravity of the Earth and Moon ; and let GS, GM, GN 



point respectively to the Sun, the Moon, and the node of the Moon's orbit 

 upon the ecliptic, so that NM is the plane of the Moon's orbit and NS 

 the ecliptic. Let MS=ca, NS = 6', NM=6, of which the first is identical 

 with the quantity denoted by the same symbol in Lecture II, but the 

 second and third are not so. 



Then, following Lecture II, the forces on the Moon are 



u. m'r . 



P+^r mMG, 



-, 3 3 cos co, in SG, 



if we ignore the parallactic terms. 

 This latter may be resolved into 



TV1 i* 



-jf- 3 cos co x (cos 6 cos 0' + sin sin & cos i) in MG, 



m'r 



m'r 



3 cos co x (sin cos 6' - cos sin ff cos i) P er Pe ndi ^ to MG in 



the plane of the orbit, 



- 3 cos co x sin 0' sin i 



perpendicular to the 

 plane of the orbit. 



