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



And if $N be the consequent increase of the longitude of the node 

 we have evidently from the figure, 



= uSt sin (N- C) cot i 



or -7 = o> sin (N - C) cot i. 



etc 



Again, the point of the ecliptic 90 in advance of N will move through 

 the space 



or (aSt cos (N C), 



perpendicular to the ecliptic, and this quantity will measure the diminution 

 in the inclination of the Moon's orbit. 



Hence we have 



8i= -to&tcos(N-C), 



or -y = to cos (N C). 



Thus we have found the rates of change of the longitude of the Moon's 

 node and of the inclination which are due to the motion of the ecliptic. 



Now, suppose the formulae which give the rates of change of the same 

 two elements, with respect to a fixed ecliptic, which are due to the Sun's 

 disturbing force, to be represented by 



r/AT" 



=-ccoBi + F(0, 0'), 



and 



dt 

 di 



dN 

 where c cos i denotes the non-periodic term in -, , c being approximately 



3 w' 2 



equal to , and F(6, 6'), f(0, 6') consist wholly of periodic terms which 



4 % 



involve the longitudes 0, 0' of the Moon and Sun respectively, as well as 

 the elements N and i. 



