516 
farther imagine a ‘‘ second fictitious sun,” so placed that the 
actual sun shall appear to be midway between this and the first 
fictitious sun; we shall then be able to describe in words the 
directions of the three disturbing forces of the second group, 
and to say that they tend respectively, for the case of our own 
satellite, to these three (real or fictitious) suns. For these 
three forces will have, for their respective expressions, the three 
corresponding terms of the development of the ¢ractor (22), 
namely, the following : 
$2,0 = § BaB (—a*)> ; * 
gir = $P'a(—a’y #5 (31) 
© 0,2 = aPaBa-' (—a)-*; 
of which the intensities are respectively 
gba; 30a; 13a"; (32) 
so that they are exactly proportional to the three whole num- 
bers, 1, 2, 5; while they are directed, respectively, to the first 
fictitious sun, the actual sun, and the second fictitious sun. 
The disturbing force of a superior planet, exerted on an infe- 
rior one, may be developed or decomposed into a series of 
groups of lesser disturbing forces, the intensities of the several 
forces in each group being constantly proportional to whole 
numbers, in an exactly similar way ; nor does the application 
of the principle and method of development thus employed 
terminate here. In the applications to the lunar theory, a and 
b, in therecent expressions, are to be regarded as denoting the 
variable distances of the sun and moon from the earth; aud 
the expressions for the forces are to be multiplied by the mass 
of the sun. Nothing depends, so far, on any smallness of ec- 
centricities or inclinations. 
V. The lunar theory is, very approximately, contained in 
the differential equation (4), provided that we regard y as the 
elliptic vector of the sun, drawn from the common centre of 
gravity of the earth and moon; and the laws of the sun’s re- 
oe ie 
