OF THE MOTION OF A SYSTEM. 211 



5x, and we have 0=m (-7— — P\ -\-m\^ (-— — P'Y or 



Py\ and in the same manner the substitution of ,..., 



' y 



for 3y, ly\ ... , gives us, for m^ (-rf— Q) + w%* 



( •J^""^'^')- • • » — ^w* ("d/3 ~^^) * ^^^"^*^ ^^^ obtain 

 Swii— 2ll|L-f + 2ffi(Py— Qa:)=:0: and by taking the 

 fluent, we have c—I.m ^ ^~"^ - ^ + S/zw {Pi/—Qx)(\l, 



[since d(xdy)z=a'd2y + dardy, and d(i/d2')=:yd2jr + dxd?/]; c 

 being a constant quantit}'. By employing the same mode 

 of reasoning with respect to the variations of a- and z, and 

 of y and z, compared together, we obtain two other similar 

 equations ; consequently 



c = Im ^^y—y^"^ + xfm {Py~Qx) d^ 

 c'-Xm ^lfZ:£!^ + 2/^ (Pz—Rx) dt, and 



= 2m yif^ + 2> (Qz-%) d^ 



Let us now suppose that the different bodies are only 

 subjected to each other's reciprocal actions, and to a force 

 directed to the origin of the coordinates. Calling the 

 reciprocal action of m and m', p, we shall have, as far as 

 this action is concerned, 0=m{Py—Qx)-\-m'{Py — QV); 



[for mP=fc£>. r.'F=PL^f^\ mQ=P^\ n^Q'= 

 P^J-Vl, as in article 322, and ,«Py + m'Fy'=^^~^ V 



J ' •^ 



P 2 



