29] DUE TO SECULAR CHANGE OF THE PLANE OF THE ECLIPTIC. 247 



Now let NM be the Moon's orbit and M the place of the Moon as 

 found from formulae in which the plane of the ecliptic is supposed to be 



L < 



fixed, and let N'M' be the Moon's orbit and M' the place of the Moon 

 at the same time taking into account the motion of the ecliptic. 



Let NM=il>, and N'M' 



Also let s denote the sine of the Moon's latitude, and ft the latitude 

 itself, in the case when the ecliptic is supposed fixed; 



And let s + 8s denote the sine of the latitude, and ft + 8ft the latitude 

 itself, when the ecliptic is supposed to be variable. 



Then s = sin i sin \jj, 



and 8s = cos i sin $81 + sin i cos 



Now let us assume that MM' is perpendicular to NM, in which case 



we shall have 



8\ji= cos i 8N, 

 and therefore 



$s cos i sin \jiSi sin i cos i cos \ji 8N, 



or substituting the values above found for 8i and 8N, 



8s = - sin ib sin (N C) . cos / cos (N C). 



c c cos i 



But if 6 denote the Moon's longitude, we have 



cos i sin t/ = cos ft sin (6 N), 

 and cos \ji = cos ft cos (6 N). 



Hence 

 8s = -^. cos ft [sin (6 - N) sin (N- C) - cos (0 - N) cos (N- <?)], 



C COS I 



or cos ft8ft = -- *"-, cos ft cos (6 - C}, 



C COS 1 



