LECTUBE XIII. 



THE MOTION OF THE APSE, AND THE CHANGE OF THE ECCENTRICITY. 



WE have seen that when the eccentricity of the Moon's orbit is not 

 considered we may write 



!=![!+ a., cos 2(0- 

 r a 



where 



H= nd-[l + h, cos 2(0- 0')], 

 3 . 2-m I 



2 1-m . Q 11 .,' 41-m' 



3 8m + - nr 



Let, us introduce the two new arbitraries e, nr by writing 

 H = hna [ 1 + /?,, cos ( 2 - 2m) 0] , 



- = ,1- [1 + 2 cos (2 - 2m) + e cos (0 - OT)], 



^ /I/ Ct 



where h is a third arbitrary, which may be chosen to suit our convenience ; 

 it must be unity when e = 0. 



dH dH dt 3 



3 r 4 



= - - mV -^-[sin (2 - 2m) + 4me cos (2 - 2m) sin (0 - CT)] 



3 A 8 a 4 



= - - mV , 2 [1 - (4a 2 + ^ 2 ) cos (2 - 2m) - 4e cos (0 - CT)] 



x [sin (2 - 2m) + 4me cos (2 - 2m) sin (0 - ra-)], 

 and also 



*- COS 2 ~ 2m ~ m% 2 ~ 2m ^ 2 sin 2 ~ 2 



