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



(2) The coefficients of the above expression will be very slightly changed 



by quantities which are proportional to 



dC 

 "dt'> 



(3) The expression for the inequality of latitude will contain extremely 



small additional terms of the form 



J* {g_ 3 sin ( - 3nt + 4n't -C)+g_ 1 sin(-nt + 2n't -C)+g 1 sin (nt - C) 



+ g, sin (3nt - 2n't - C)}; 



that is to say, these terms will involve the sines instead of the cosines of 

 the same arguments as before, and the coefficients of these new terms are 

 proportional to 



da) 



Hi- 

 ll. Theoretical explanation of- the same inequality, which was originally 

 given, in substance, in Godfray's Elementary Treatise on the Lunar Theory. 



The general principle of this explanation may be very simply stated. 



If, for a moment, we suppose the plane of the Moon's orbit to remain 

 fixed, and imagine the plane of the ecliptic to turn through a very small 

 given angle about a line in its own plane, this will give rise to cor- 

 responding small changes in the longitude of the Moon's node and in the 

 inclination of the orbit to the ecliptic, and the magnitude of these changes 

 will depend on the angular distance of the Moon's node from the line 

 about which the ecliptic is supposed to be turning. 



If now the planes of both orbits be supposed to vary continuously, 

 the total changes in the longitude of the node and inclination of the orbit 

 produced in an indefinitely small time will be found by adding together 

 the changes respectively due to the motion of the plane of the ecliptic, 

 and to the motion of the plane of the Moon's orbit with respect to the 

 ecliptic when the latter is supposed to remain fixed during that small 

 time. The motion last mentioned is given by the formulae of the ordinary 

 Lunar Theory, in terms of the disturbing force of the Sun. In consequence 

 of the action of this force, the Moon's node gradually makes complete 

 revolutions with respect to the line about which the ecliptic is turning, 

 and the summation of all the momentary changes of node and inclination 

 due to the motion of the ecliptic will produce periodic changes in those 



