Motions of a System of Bodies. 41 



duced to the same axis. Hence — wi' p' is one of the forces which 



compose P in the first of (I), and m -p' is one of the components of 



P' in the second of (1) ; but these forces may be made to disappear 



by multiplying the first of (1) by m, the second by m', and then 



adding the products. It is hence evident that if the first of (1) is 



multiplied by m, the second by m', and so on, and the products add- 



d'^x d'x' 

 ed, there will result m--T7^-{-m'-T-Y-{-^c, = mP-\-m''P'-\-hc. which is 



independent of any actions of the bodies on each other; for the 

 terms arising from the reciprocal actions of every two of them 

 will destroy each other as above : in the Same way (2) and (3) give 



//2 flf w3 y/ d^ z d' z^ 



^dP +"*' J^7+^c.=mQ4-m'Q'+&;c., ^^+"^'^+ &^c. =mR 



+m'R'+&;c. ; put ?n+m' + &£C.=M, mxi-m'x' -\- he. =MX, my-\- 

 my+&c.=MY, mz+m'z' +hc. = MZ, and let mP+m'P'+ &;c. 

 be denoted by SmP, mQ,'{-m'Q,'-\-k,c. by SnQ, wR+w'R'-f&ic. by 

 SmK ; then by substitution and reduction the above equations become 

 d'^X SmP d'Y SmQ, d'Z SmR 

 'dF = 'W' W^=lt' ■rf^^"ir(4)5 which are mdepen- 



dent of any terms arising from the actions of the bodies on each oth- 

 er. X, Y, Z, are evidently the coordinates of the center of gravity 

 of the system ; and it is manifest by (4) that the center of gravity 

 moves in the same manner that the unit of masses would do if it was 



, , ^ SmP SmQ SmR 



collected at the center, and acted on by the forces ^, , — irj-, t^ - , 



in the directions of the axes of x,y,z, respectively. 



If the bodies are subjected to no forces but their mutual actions, 



d-'X 

 then as shown above SmP = 0, SmQ=0, S/nR=0 ; .'. — jf— =0, 



d'Y d'Z dX dY dZ 



W^^' 'df^^'^ ^^°'^ ^"^'S''"'' S'^" "57=^' ^7=^'' rf7==^"' 



(5); V, V, Y", being the arbitrary constants, they also express 

 the velocities of the center of gravity in the directions of the axes of 

 X, y, z, severally, which are hence constant; and it is easy to see that 

 the motion of the center is rectilineal and uniform, unless V, V, V 

 are each =0, in which case the center is at rest. The equations 

 (4) are easily adapted to the motion of a solid by putting M= to its 

 mass, and denoting any indefinite element of it by dM ; and repre- 

 senting indefinitely by P, Q, R, the forces which act on dM in the 

 Vol. XXIV.— No. 1. 6 



