M. Poinsot on the Percussion of Bodies. 267 



46. If we suppose j^ 



the two roots of the equation will be equal and have the value 



cv = ih. 

 In this particular case of the movement of the body, therefore, 

 there is only one point which can be a centre of perfect reflexion, 

 and its distance from the centre of gravity is one-fourth of that 

 of the latter from the centre of percussion. 



Centres of no ueflexion. 



47. To find the points by means of which the body suffers no 

 reflexion after the shock, we have only to set 



u'=0, 

 and therefore 



or, substituting a6 for ii, and k for ■— , 



whence we deduce 



x=0 and w=h, 

 two values which correspond respectively to the centre of gravity 

 and centre of percussion. It is evident, in fact, that if an 

 obstacle be presented at either of these centres, the velocity of 

 translation will be destroyed. The only difference between the 

 two cases is, that in the first the velocity of translation u is alone 

 destroyed, without that of rotation being altered, whereas in 

 the second case both u and 6 are destroyed. 



48. In order to find the point where an obstacle must be pre- 

 sented in order that the velocity of translation may suffer no 

 alteration, we should merely have to make m'=u, which would 

 lead us to the value 



x= —a, 



corresponding to the spontaneous centre of rotation. In fact 

 since this point of the body is at rest each instant, it is clear 

 that it cannot strike an obstacle there presented, and on that 

 account the motion yf the body can suffer no alteration. 



General Remark. 



49. We may here remark that the theory of the centres of 

 reflexion is, in reality, the same as that of the centres of percus- 

 sion ; for the relation 



Q+i^=P, 



which exists between the two components Q and p, and the 

 original force V, shows at once that the same point which is the 



