517 



lative elliptic motion, with respect to that centre of gravity, 

 are then contained in the following differential equation, which 

 takes the place of the equation (5) : 



d?=77(^T' ^ ^ 



Indeed, when we come to consider the small disturbing 

 forces which belong to the second group, and which depend on 

 the inverse fourth power of the sun's distance, the correspond- 

 ing terms of the development of the first member of the formula 

 (6) are then, for greater accuracy, to be multiplied by the 



fraction — ■,, which expresses the ratio of the difference to 



m+m' 



the sum of the masses of the earth and moon. But if we neglect, 

 for the present, those small disturbing terms, we may regard 

 that formula (6) as accurate, without as yet neglecting anything 

 on account of smallness of eccentricities or of inclinations ; and 

 even without assuming any knowledge of the smallness of the 

 moon's mass, as compared with the mass of the earth ; y still 

 denoting, as just stated, the elliptic vector of the sun. And 

 thus, if the moon's geocentric vector j3 be changed to the sum 

 /3 + 8j3, where the term 8/3 is supposed to depend on the dis- 

 turbing force, and to give a product which may be neglected 

 when it is multiplied by or into the expression for that force, 

 we shall have the following approximate differential equation, 

 by developing the disturbed or altered tractor ^(j3 + 8/3), and 

 confining ourselves to the first power of g/3 : 

 d^gjS d^^ _ ^ + m' _ 



"dF + d«-^ iiVi-fi')' 

 ^:l.(Sp + 3p-SpP) + -^,0 + 3,-.p,). (34) 



The disturbance g|3 of the moon's geocentric vector is thus 

 exhibited as giving rise to an alteration d<p((i) in the corres- 

 ponding tractor 0(j3), which alteration is analogous to a dis- 



