lvi 
if then we reject the terms of the same order as m” 3” y~*, 
that is, terms depending on the inverse fourth power of the 
distance of the sun from the earth, the disturbing part of the 
expression for the second differential coefficient, taken with 
respect to the time, of the moon’s geocentric vector, will 
reduce itself in this notation to the following : 
—wy// he —I Cit" 3 , 
Te ry = 184 37-8) 1D 
This symbolic result admits of a simple geometrical in- 
terpretation. The symbol y—'Py denotes a vector in the | 
plane of the two vectors 3 and y, which has the same length 
as (3, and is inclined at the same angle to y, but at the other 
side of that line; so that y bisects the angle between § and 
y'By. If then we conceive a fictitious moon among the 
stars, so situated that either the sun, or a point opposite to the 
sun, as seen from the earth, bisects the are of a great circle on 
the celestial sphere, between the fictitious and the actual moon 
(the bodies being here treated as points); and if we decom- 
pose the sun’s disturbing force on the moon into two others, 
directed respectively towards the extremities of that celestial 
are which is in this manner bisected: one component force, 
resulting from this decomposition, will be constantly ablati- 
tious, tending directly to increase the distance of the moon 
from the earth, and bearing to the attractive force in the 
moon’s undisturbed relative orbit, a ratio compounded of the 
direct ratio of half the mass of the sun to the sum of the masses 
of the earth and moon, and of the inverse ratio of the cubes 
of the distances of the sun and moon from the earth; and the 
other component force, directed towards the fictitious moon, 
will be exactly triple of the ablatitious force thus determined ; 
provided that we still neglect all terms depending on the 
inverse fourth power of the sun’s distance, as we have done in 
deducing the equation (17), of which the theorem here enun- 
ciated is an interpretation. A similar result, of course, holds 
good, for every satellite disturbed by the central body of 
