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



and substituting these values in the 2nd and 3rd equations, they become 



0'33322,76c_ 1 + 0-01261,74 c, + 0'01597,lw = 0, 

 0-01261, 74c_ 1 + 0-00982,925c, + 2-15975, lft> = 0. 



Whence again, we find 



c_,^ 8-69441&), 



c, = -230-8866 o>, 



from which by substitution we obtain 



c_ 3 = 0-01097&), 

 c a = - 0-349 15w, 

 e. = - O'OO 136w. 



Hence the solution of the differential equation for Sz is 



Sz = &) {0-01097 cos ( - 3nt + kn't] + 8'69441 cos ( - nt + 2n't) - 230'8866 cos nt 



- 0-3491 5 cos (3t- 2n't) - 0'00136 cos (5nt - 4n')}. 



Here &> is expressed in terms of the circular measure, and Sz in terms 

 of the unit of length denned before. 



If s denote the sine of the Moon's latitude, 



and if Ss be the change in * due to the secular change in the plane of 

 the ecliptic, we have 



since Sr = 0, 



according to the suppositions made above. 



Also 

 \ = 1-00090,74 + 0-00718,65 cos 2 (nt - n't) + 0'00004,58 cos 4 (nt - n't). 



Hence by substitution 

 82 = w {0-0369 cos ( - 3nt + 4n't) + 7 '8727 cos (-nt + 2n't) - 231 '0661 cosnt 



- T1789 cos (3nt - 2n't) - 0'0079 cos (5nt - 4n'*)]. 



