Ivi 



if then we reject the terms of the same order as m" j3^ -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 : 



;;^C^;;^f --)=i-«(-M/3+3.-/3.). (,r) 



This symbolic result admits of a simple geometrical in- 

 terpretation. The symbol 7""^ /By denotes a vector in the 

 plane of the two vectors j3 and y, which has the same length 

 as j3, and is inclined at the same angle to 7, but at the other 

 side of that line ; so that 7 bisects the angle between j3 and 

 y-'jSy. 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 arc 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 

 arc 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 



