168 M. Poinsot on the Percussion of Bodies. 



distances of the centre of gravity, and of the ordinary centre of 

 percussion from the same point O. 



But to arrive at this theorem, let us first seek the percussion 

 which the rotating body would produce at a fixed point C, situ- 

 ated at any distance x from the ceiitre of gravity G. Afterwards 

 we shall see what value of x w'ould render this percussion a 

 maximum. 



15. At the moment of impact we may regard the body as if, 

 after having been previously in a state of repose, it was suddenly 

 acted upon by a single force P applied at the point C. We 



may always conceive this force P, too, to be decomposed into two 

 other parallel forces ; the one, Q, being applied at the point C, 

 and the other, p, at 0', the reciprocal of C ; that is to say, at 

 the point which would correspond, as spontaneous centre of rota- 

 tion, to the point C considered as a centre of percussion. The 

 point C being at the distance x from G, the point 0' is on the 



other side at the distance — . The distance between the compo- 



nents Q and p, therefore, '\% x-\ ; that between P and p is 



h-\ : hence by the theory of parallel forces* we have, for the 



component Q applied at C, the value 



and for the other component p applied at 0', the value 



x^—hx 



J9 = P, 



K2 + A-2' 



Now of these two components, the latter p which strikes at 0' 

 can produce no percussion at the fixed point C, since this point 

 C is the spontaneous centre with respect to the point 0' where 

 the force jt» is applied. The only force which remains to strike 

 the obstacle C, therefore, is the component Q, which is directly 

 applied to that point; whence it follows that the percussion 

 produced by the body against the fixed point C, placed at the 

 distance x from the centre of gravity, is exactly expressed by the 

 function 



^_p K^ + lix 



'*' Poiusot's EMments de Statique, art. 28. 



