M. Poinsot on the Percussion of Bodies. 165 



Corollaries. 



8. The equation 



which gives the distance of the point from the centre of gra- 

 vity G, shows that the position of the spontaneous centre depends 

 neither upon the mass M of the body, nor upon the intensity P 

 of the impulse apphed at the centre of percussion C, but solely 

 upon the distance h between this point C and the centre of gra- 

 vity G. The product ah having always the same value, it follows 

 that a increases in the same proportion as h decreases ; thus the 

 nearer the centre of percussion approaches the centre of gravity, 

 the more the spontaneous centre on the other side recedes from 

 the same, and vice versa. 



9. If we suppose /i = 0, we find a = cc . Hence one might say 

 that when the centre of percussion coincides with the centre of 

 gravity, the spontaneous centre is infinitely distant : but, strictly 

 speaking, there is then no spontaneous centre of rotation; for, 

 since the body, in this case, receives a shock at its centre of 

 gravity, the only effect will be a simple translation in space. 



10. On the other hand, if we suppose h = (X> , we find a = 0; 

 that is to say, the spontaneous centre coincides with the centre 

 of gravity. But, strictly speaking, there is no longer a centre of 

 percussion in this case ; for if the body turns around its own cen- 

 tre of gravity, which remains fixed in space, the percussion P, 

 which set it in motion, — no matter at what point we suppose the 

 same to be applied, — must necessarily be zero, otherwise the centre 

 of gravity would not be motionless, but would have a finite velocity 



P 



=rj. Thus there is no simple and finite percussion P which can 



give rise to a spontaneous centre coincident with the body^s 

 centre of gravity. 



All that can be said is, that if the centre of percussion is in- 

 finitely distant from the centre of gravity, the spontaneous centre 

 is infinitely near to the same ; but in dynamics it does not fol- 

 low from this that these two points of the body can ever coincide, 



P 



for the one G will have the finite velocity ^, whilst the velocity 



of the point is from its very nature zero. 



In one way only can we imagine the centre of gravity and the 

 spontaneous centre strictly to coincide ; it is by supposing P = 0, 

 /t = x , and the product or moment Vh equal to a finite quantity. 

 Under this mathematical hypothesis, the body would turn around 

 its centre in virtue of the finite moment P/t, and this centre 

 would remain at rest in consequence of P = 0. It must be con- 

 fessed, however, that a percussion zero applied at an infinite 



