'238 M. Poiusot on the Percussion of Bodies. 



This latter centre is infinitely near the fixed point 0, and this 

 percussion is infinitely great. 



15. In a former article* we found that the point hy means 

 of which a body ]M can communicate the greatest possible velo- 

 city to a free point of the mass m, before at rest, is not the centre 

 of greatest percussion of the same body against a fixed point, 

 but that it is a new point situated at a distance X from the spon- 

 taneous centre of rotation of the body M expressed by 



X=±^„= + K<l + M), 



where K is the arm of inertia of the body j\I with respect to its 

 centre of gravity G, and a the distance of this centre G from the 

 spontaneous centre of rotation 0. If, in place of the simple 

 body M, we consider the system M+yu. composed of M and a 

 material point //, placed at I, we must change, in the preceding 

 expression for X, JNI into M + ft, K- into K'^, and a into a'f, 

 K' being the arm of inertia of the system with respect to its 

 centre of gravity ; in this manner we find 



\ m J 



or 



m 



If we now put [ji-=QC , in order to pass to the hypothesis of a fixed 



point at I, around which the simple body M turns, we shall have 



K'=0, «' = 0, (M + /t)K'2=M(K2+<i2)^ 



where K is the arm of inertia of the simple body M around its 

 centre of gravity G, and d the distance of this centre from the 

 fixed point I ; ;^I(K* + fi^) is clearly the moment of inertia of M 

 with respect to the fixed point I. 



Thus when a body M turns around a fixed point I, the centre 

 V of greatest velocity communicable to a free point m is at a di- 

 stance from I equal to 



IV=a/M2!±?). 

 V m 



This point V, therefore, depends upon the ratio which exists be- 

 tween the mass M of the striking body and the mass m of the 



' * Chap. I. art. 30. 

 K'- 

 t Here c'= —rr> "here ^' denotes the finite distance of the centre of per- 



h 



cussion fi'om the centre of gravity of M+/n. 



