21] ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. 143 



In the present communication, however, I shall confine my attention to the 

 principal term of the change thus produced in the acceleration of the Moon's 

 motion, deferring to another, though I hope not a distant, opportunity, the 

 fuller development of this subject, as well as the consideration of the secular 

 variations of the other elements of the Moon's orbit arising from the same 

 cause. 



In what follows, the notation, except when otherwise explained, is the 

 same as that of Damoiseau's Theorie de la Lune. 



6. If we suppose the Moon to move in the plane of the ecliptic, 

 and also neglect the terms depending on the Sun's parallax, the differential 

 equations of the Moon's motion become 



d"u 1 m'u' 3 



2 V dv 



dt 13 



T = *, + o Z.Y 

 dv fm 2 h 



s . n _ _ 3m' / dru\ fu^dv _ 



K \ dv*j } u' 



m! (u' 3 dv . . ,. 27 m' 2 f fu'^lv . . , A T 



Z.Y 2 T- sm ( 2v ~ 2v ) + T- Til -*- sin ( 2 " ~ 2v ) 

 hit') u* 8 W |_J u J 



In the solution usually given of these equations, u is expressed by 

 means of a constant part and a series involving cosines of angles composed 

 of multiples of 2v 2mv, cv ra-, and c'mv nr' ; also t is expressed by means 

 of a part proportional to v and a series involving sines of the same angles ; 

 the coefficients of the periodic terms being functions of m, e and e'. Now 

 if e' be a constant quantity, this is the true form of the solution, but if 

 d be variable, it is impossible to satisfy the differential equations without 

 adding to the expression for u a series of small supplementary terms de- 

 pending on the sines of the angles whose cosines are already involved in 

 it, and to that for t, similar terms depending on the cosines of the same 



dd 



angles, the coefficients of these new terms involving -y - as a factor. 



fu' 3 dv 



The quantity I sin (2v 2v'), which occurs in the above equations, 

 J u 



is proportional to the variable part of the square of the areal velocity, 

 and consists, in the ordinary theory, of a series of periodic terms involving 

 cosines of the angles above mentioned. In consequence, however, of the 

 existence of the new terms just described, there will be added to it a 



