236 NOTE ON THE INEQUALITY IN THE MOON'S LATITUDE [29 



and therefore 



d 2 -n d^ . &y dz 



3? cos wt + df sin wt = 3? ~ 2w eft ~ "*' 



d-t, d*<n . <frz dt/ 



~ cos tat j^ sin (at -, + 2w f - wrz. 



df dt dt- dt 



Now substitute for r), and ', tj', ' their values in terms of 

 x, y, z and x', y' respectively, in 



(1). 



(2) cos w + (3) sin w<, 



(3) cos cat (2) sin (at, 

 bearing in mind that 



since each of these quantities represents rr'cos(r, ?'), and we have 



d"jc u.x m'x 3m'x' 



+**- s + 



r' 5 



py _ m'y 3m' y' 

 dt~ "dt" y ^l* = ' r" ' r'^ 



drz %> dy ^ piz _ in'z 



which are the equations of the Moon's motion, with reference to the variable 

 ecliptic. 



The motion of the ecliptic is so slow (that is, w is so small) that 

 the terms involving ar may be neglected. 



We will now change the notation by writing for the Moon's coordinates 

 x + Sx, y + &y, and z + 8z, instead of x, y, z respectively, in which expressions 

 the new quantities x, y, z are taken so as to satisfy the equations of motion 



d*x fj.x _ m'x 3m' xf . 

 tfy 



T i t*. ' *T ~ 



1 



cty uy m'y 3 m'y' . . ,, 



^ + * = - ^ + -J- ( xx + yy}' 



mz 



