242 M. Poinsot on the Percussion of Bodies. 



the same as that which the point C of a system composed of M 

 and in would assume when struck in C by a force mv which 

 ]iassed entirely into this system. But this velocity is easily 

 found. 



In fact let g be the centre of gravity of the system composed 

 of ]\I and m, x the distance CG, and n the ratio m : M ; the lines 

 yG and ^C will be thus expressed : 



Let MK* represent the moment of inertia of the body M with 

 respect to the principal axis under consideration passing through 

 G, and (M + h;)K'- be that of the system with respect to a 

 ])arallel axis through its centre y ; then, as is well known^ 



(M + m)K'^=MK^ + M(^-^)V ra{^^\ 

 whence 



^^~- (« + ir • 

 Further^ the force mv applied to the system M + m at the distance 



-. from its centre of gravity g imparts to the system, first, a 



velocity :^ or ^ common to all its points ; and secondly, 



a rotation around g with an angular velocity 6 such that 



This gives for 6 the value 



f._ nvx 

 (n + l)2K'2' 



and for the velocity of the point C in consequence of this rota- 

 tion 6, the value 



nvx'^ 



Tliese two velocities of the point C being like -directed, it8 

 total velocity u will clearly be 



+ ■ 



or, substituting the above value of K'^, 



whence m'C deduce for the quantity of motion which is imparted 



