OF THE MOTION OF A SYSTEM. 225 



substitutiDg 8X+Sx^ for 8a?, we have [0 = Sw (gZ + gj,) 

 (^_ p)+... but 2m8X(^ — P) = 0, because 



2^71 /!— — P \ =0, ^X being the same for all the quanti- 

 ties of which the sum is denoted by 2 ; consequently 0= 

 2m Sa:,(-^ — P ) . . . , or, d^x being equal to d^jj 0=2m 



an equation which is exactly of the same form with the 

 equation (P), 'supposing the forces P, Q, R, to depend 

 only on the coordinates ^^, 3/^, z^, /^, .... If, therefore, 

 we apply to it the same reasoning, as was grounded on that 

 equation, we may derive from it the same conclusions, with 

 respect to the preservation of the impetus, aud the de- 

 scription of areas, relative to the moveable origin of the 

 coordinates. 



If the system is not subjected to the action of any ex- 

 traneous force, its centre of gravity will have a rectilinear 

 and uniform motion (322) : so that if we suppose the or- 

 dinates jr, y, and z to begin at the centre of gravity, the 

 laws in question will always we observed: and X, Y, and 

 Z being the coordinates of the centre of gravity, we shall 

 have, by the nature of this point, 0:=z'I,mx^y OzzT/w^^^, and 



^ ^dy-^ydx ZdF— FdZ ^ 



0— Smz : whence we have 2lm — ~-f zz r: • ^m 



' dr df 



^ Tdy,— ydx, , ^ da^ + d^^ + dz^ 



+ ^^ d/ - ^"^ ^^ " tt- = 



dX'^+dr^+dzy ^_, di-;+dy2+ds/_ . ,^ . 



j^^ 2m + 2w2— ^ ^^ ^[, for dX and 



^Y being common to all the system, we have 2wdX= 

 dXSm, SmdF^dYSw; and the sums of the squares of 



