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 

 ihree/orces will have, for their respective expressions, the three 

 corresponding terms of the development of the tractor (22), 

 namely, the following : 



^2,o=f/3a/3(-a2)-i; "I * 



^,,,z:fi3^a(-ar^; (31) 



.0„,2i=^ai3ai3a-n-a^)-^; J 



of which the intensities are respectively 



f6^-a-*; ib'a-'; lAb'^a-'; (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 

 h, in the recent expressions, are to be regarded as denoting the 

 variable distances of the sun and moon from the earth ; and 

 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- 



