Iv 



Consider a system of three attracting masses, m, w/', m", 

 with their corresponding vectors, a, a', a"; and make for 

 abridgment a! — a = j3, and a" — a = y ; we shall have, 

 by (1), for the differential equations of motion of these three 

 masses, referred to an arbitrary origin of vectors, the following : 

 d'^a m' m" l 



dF--/3v/(-i3^-) "^ -Tv/(-"7)' 

 d^(a 4- /3) _ m %_ 



~~dF~ - WVG-~W) "^ (i3-T) V{-(i3-7n '"^^^^ 



<\\a 4- 7) _ »i *n' 



which give, for the internal or relative motions of in' and m ' 

 about tn, the equations : ^ 



d-/3 m + 7n' „( O-7)-' 7-' 



f (p-7) , T~' » 



\V[-(/3-7)n W(-7-')/ 



d/^ -/3v^(-/3^)^ \V[-(/3-7)n ^(-7-')/' 

 d^y _ m + m" / (7-/3)-' , /3-^ 1 



dt' ~lV(-y') ^ W{-(7-/3)n V(-/3^)i • 



If we suppress the terms multiplied by m" in the first of 

 these equations (14), or the terms multiplied by in' in the 

 second of those equations, we get the differential equation of 

 motion of a binary system, under a form, from which it was 

 shown to the Academy last summer, that the laws of Kepler 

 can be deduced. If we take account of the terms thus sup- 

 pressed, we have, at least in theory, the means of obtaining the 

 perturbations. 



Let 771 be the earth, H^' the moon, 7n" the sun ; then )3 

 and 7 will be the geocentric vectors of the moon and sun ; and 

 the laws of the disturbed motion of our satellite will be con- 

 tained in the two equations (14), but especially in the first of 

 these equations. By the principles of the present calculus we 

 have the developments, 



(7 - (5)-' = 7-' 4- 7-737-' + 7-' /37-' /37-' + . . , (15) 

 and 



V(-y') _f, /37 + 7/3 , /3'^)-i_, ./37+7^^ .,,^, 



