240 CELESTIAL MECHANICS. I. vii. 25. 



their importance in the system of the world makes it con- 

 venient to develope them somewhat more in detail. 



If the coordinates of the particle Dtw, referred to the 

 centre of gravity, be y, y , z', so that its whole motion is 

 determined by the sums jr + x', y-\-y andz + z'; "the 

 forces destroyed at each instant in the particle Dm, in the 

 respective directions, considering the element of the time 

 as constant, will be 

 ddx + ddo/ 



d^ 



nm + PditBm ; 



— — ^. ^ DOT + Qd^DOT ; and 



d^ 



dds + ddz' . r>j^ 



nm -\- KatDm. 



dt 



It is therefore necessary that all the forces thus destroyed 

 should be in equilibrium with each other" [that is, as 

 causes and effects] : and that the sum of all the forces pa- 

 rallel to any given axis, should vanish (307) : hence we 

 have the three following equations 



t> Dm=zSPDOT, S — '■^— — ■- Dtnz=:SQDm; 



dt dt 



and S dot==:S12dot. Now since t, y, and z 



dt 



are the same for all the particles, they may be excluded 



from the quantity under the sign S ; so that we have 



g __^ vm—m -— , . . . ; we have also, by the nature of 



the centre of gravity Sa:'Dm=:0, . . . ; consequently S 



~Dm=0,S ^'d7w=0, and S^^ dm=0 : and lastly 



^dd^^g ^^^SQDm,^ndm^^ = S RDm, 



dt^ dt^ dt^ 



These three equations determine the motion of the centre 



