29] DUE TO SECULAR CHANGE OF THE PLANE OP THE ECLIPTIC. 243 



elements, the magnitudes of which, at any given time; like the momentary 

 changes themselves, will depend on the angular distance, at that time, 

 between the Moon's node and the line about which the ecliptic is turning. 



The combined effect of these periodic changes in the position of the 

 node and in the inclination is to produce the inequality in latitude which 

 is now under consideration. 



The motion of the Moon's node is not uniform, but the principal in- 

 equalities by which that motion is affected have periods which are short 

 compared with the time of revolution of the node. 



Hence the periodic changes of node and inclination above described, 

 will be accompanied by others which are due to the same cause, but which 

 in consequence of the shortness of their periods will be comparatively un- 

 important, and the combined effect of these changes in the elements will 

 be to add other terms which are equally unimportant to the expression 

 of the inequality in latitude. 



We proceed to find the analytical expressions for the changes in the 

 longitude of the Moon's node and in the inclination of the orbit, due to 

 the motion of the plane of the ecliptic, supposing the Moon's orbit itself 

 to remain fixed. 



Take C the longitude of the instantaneous axis about which the ecliptic 

 is rotating at the time t, 



a the angular velocity of the ecliptic, 

 N the longitude of the Moon's node, 

 and i the inclination of the orbit, at the same instant. 



Then, in the indefinitely small time St, a point of the ecliptic situated 

 in any arbitrary longitude L will move through an angular space 



d)St sin (L C) 

 in a direction perpendicular to the ecliptic. 



Hence the point of the ecliptic originally coincident with the node N 

 will move through the space 



perpendicular to the ecliptic. 



312 



