OF THE moon’s MEAN MOTION. 
399 
of the areal velocity will be of the order of the square of the disturbing force multi- 
plied by the rate of change of the earth’s eccentricity. 
It is evident that the amount of the acceleration of the moon's mean motion will 
be directly affected by this alteration of areal velocity. 
5. Having thus brieflyindicated the way in which the effect now treated of originates, 
I will proceed with the analytical investigation of its amount.' 
In the present communication, however, 1 shall confine my attention to the prin- 
cipal 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 “ 7’heorie de la Lime.” 
6. If we suppose the moon to move in the plane of the ecliptic, and also neglect the 
ternjs depending on the sun’s parallax, the differential equations of the moon's motion 
become 
1 m'u'^ 3 m'u'^ 
cos (2^ -2d) 
3 m'u'^ du . 3m'/ 
2 4V 
d^u \ Cu'^dv 
dt 1,3 m! Cu'^dv . ,27 m'^ 
dv—hu^~^2 
Cu'^dv 
sin ( 
sin (2t/ — 2d) 
2v—2d) \ 
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 
2i>—2mv, a— CT, and c'mv-vs ' also t is expressed by means of a part proportional 
to » and a series involving sines of the same angles ; the coefficients of the periodic 
teims being functions of w, e and e'. Now if e' be a constant quantity, this is the 
true form of the solution, but if e' be variable, it is impossible to satisfy the differen- 
tial equations without adding to the expression for u a series of small supplementary 
terms depending 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 angles, the' 
coefficients of these new terms involving ~ as a factor. 
CU'^dv 
The quantity sin {2v—2v), which occurs in the above equations, 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 series of small terms involving sines of the same angles, together 
with a non-periodic part of the form ^He’de' or \ The introduction of this term 
3 G 
MDCCCLIII. 
