OF THE MOTION OP A SYSTEM. 227 



For since £;» -^^ C -^ 2i:fmm' Fdf, (319) 



^^"^ — df^ — "^''^ dZ^^ "^ • • • ' '^"^tip'y- 



ing by 2w=»j + wi' + m" + . . . , we have {m + wi -{-...) {m 

 di-^ + di/^ + dz^ ,dx'2-fd?/2-rd2'2 , ^, ^^ ^r . 



JPd/*; but the first member of the equation of the propo- 

 sition will be found to be equal, when expanded, to the first 

 member of this last, the difference of each part becoming 

 = : thus, taking m and m for an example, the difference 

 will be mm' (dx'—dxy—(m + m') (mdx^ H- wz'd j;'^) — mm'dx'^ 

 + mm dar^ — 2mm'dx'dx — mmdx'^—mnidx'^ — mm'dx^ — m'm* 

 dx'* zz—2mm'dx'dx—mmdx^ --rdrridx'^— — (mdx + m'dxY; 

 and by the successive addition of the different pairs, this 

 difference will become (Imdxf^zO, since ^mxzuO and 

 Xmdx—Of by the properties of tbe centre of gravity, con- 

 sequently] the two expressions are equal for the whole 

 system. 



§ 23. Principle of the least action. Combined with 

 that of the preservation of impetus, it gives the general 

 equation of motion. P. 63. 



334. Theorem. The momenta of a sys- 

 tem of bodies being multiplied by the fluxions 

 of the spaces respectively described, the sum 

 of the fluents, taken, for the whole system, 

 between any given points of space, is always 

 a minimum. 



The equation (R) (319), lmv^zzc + 2(p=2Xmf (Pdx-h 

 Qdy-fJRdz), affords us tbe variation SwvSu^Sw (P^x-t- 

 Qdy-^R^z), and combining this with the equation (P)(317) 



Q 2 



