Variations of the Arbitrary Constants in Elliptic Motion. 249 



cate to the attracted bodies in the time dt ; .'. — dt is the velo- 

 city with which M' and m approach each other at the end of the 

 instant dt, by their mutual attractions during the same time, and 



— denotes the force of attraction of m towards M' regarded as 



at rest. 



Put M'-{-m = M, then resolving the attractions in the direction of 



M-X m'x' m'{x' — x) 

 the axis of x, we have —^ + -jj- — ,/- — for the whole force 



with which m is attracted in that direction towards M regarded as 



m'x' m'{x' — x) 

 at rest, for —77 draws M' towards m and ,jt^ — draws m from 



M^, and therefore has been written with the sign minus ; in the 



My m'y' m'(y' — y) Mz m'z' m'{z' — z) 



same way, we have -^+-;7i- - — ^ITTs— ' 7F+ ^TT" — ~tTz 



for the whole forces of attraction of m towards M' in the directions 



of the axes of y and z ; and by multiplying these attractions by dt 



we shall have the velocities which they will severally communicate 



to m towards M' in the same directions. 



dx dy dz 

 Now -TT') TT' -77 denote the velocities of m in the directions of 



X, y, z severally at the time t, and we shall suppose that they tend 

 to increase the coordinates ; it is evident, that at the end of the time 



dx dx dy dy dz dz 

 dt, the velocities will become j7+^j7' 77~^^'Jf' ^7"^"^^'^^^^^ 



are evidently less than the former velocities since m is attracted 



dx dy dz 



towards M', hence we shall have —d-rj — ^j7' ~~dl7' ^°^ ^^^ 



velocities received by m in the directions of x, y, z severally in the 



time dt; but the velocities communicated and received in the same 



directions in the instant dt, are evidently equal to each other, .*. by 



d^x M.X m'x' m'{x' — x) d^y 



making dt const, we have ^+77 + ^ -"^771 =0? jj^ + 



Ml/ m'y' rn'{y' — y) d^z M-Z m'z' m'[z' — z) 



Multiply the forces in the directions of x, y, z by dx, dy, dz sev- 

 erally, and put — ^Q for the sum of the products, then by taking the 



Vol. XXX— .No. 2. 32 



