OF THE MOTION OF A SYSTEM. 231 



those of the preservation of impetus, the coDstanoy of 

 areas, and the motion of the centre of gravity. By sub- 

 stituting da:, dy, and dz, for 3ar, Sy, and 3jr, we obtain 



..{d.d(^^.f )^d,d(|±) +dzd(y/-) =..„d„ 



d<?T=Smt;dpd/: for. dxd (^.f ) = ^ df + i d^d ^' 



vd* u' at V V Qt 



and the three parts together make --— d h — d ^-r- 



° dt V V 2dt 



= ^ (— • Tt) — -^vdvdtzz d { ^i;«dO — ^di;d* = d 

 ^ V dt ^ V ^ V ' 



/<pvdt) — (pdvdtzzvd(pdt, and the integral is truly ex- 

 pressed by Xmvdpdt.] Hence, dividing by dt, and taking 

 the fluents, S/mrd^ = c + Xjm {Pdx+Qdy + 12dz); or, 

 supposing the latter member an exact fluxion, and equal 

 to dx, we have the equation i 



'2fmvd(p=zc+x (T) 



an equation resembling (J?) (319) and which is converted 

 into it by making (pzzv; consequently the principle of liv- 

 ing force is maintained in this hypothesis, if we understand 

 by living force the product of the mass into " twice" the 

 fluent of the velocity multiplied by the fluxion of tiiat 

 function of the velocity, which expresses the force. 



336. Theorem. The sum of the finite 

 forces of a system, reduced to any given di- 

 rection, is constant, and vanishes in the case 

 of equihbrium. 



If we substitute, in the equation {S), ^x-\-^x^^ for 3^, 

 hj + ^y\ for gy', dz + h\ for gz; ^x-hdaf\ for ^x" , . . . ; we 

 I shall obtain, by making the three variations vanish sepa- 



