Variations of the Arbitrary Constants in Elliptic Motion, 249 



M'+m 

 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 



M'-f-m 



at rest. 



M'reg 



Put M / +»»=M, then resolving the attractions in the direction of 



, . Ma? m'x' m / (x / — x) 



the axis of x, we have —7 + -77 — — -,,- — 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 rri — draws m from 



r 3 r 



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



• My roV m'(y' — y) Hz m'z' m'(z' —z) 



same way, we have ~^+-~ - ^»7 ^ 7T+ ^^—^ 



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 



dt 9 di } dt ^ enote 4 ^ e velocities of m in the directions of 



#, y, z severally at the time t y 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 

 of, the velocities will become j7-\-d :j7' ,/7+^j7' ^7 + 0-77? which 



are evidently less than the former velocities since m is attracted 



dx dy dz 



towards M', hence we shall have — d-r> "^di y ~^dt' 



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 2 x Mx m'x' m'(x'—x) d*y 



making dt const, we have ^+77 + ^77 — — ">"i =0 > dF * 



My my m'(y-y) i*z M* rn^ m'(z'-z) 



ra+^a — r//3 — u > dt 2 '*' r 3 r' 3 if'* ~ v > \*r 



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

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



Vol. XXX— .No. 2. 32 



