270 M. Poinsot on the Percussion of Bodies. 



passing at the distance x from its centre of gravity, the rotation 

 6 of the body is changed to another 6', such that 



Hence to find the point which cori-esponds to a maximum of 6\ 

 that is to say, the centre of maximum conversion, we have merely 

 to make 



dx ' 

 we thus obtain the quadratic equation 



ux'^-2Y^Wx-uK?==0, 



whence, on replacing ?< by its value a9 or -j-d} we deduce 

 x = h± \/rf+K^; 



both these roots are real, one being positive and the other 

 negative. 



Thus there arc always two centres of maximum conversion ; 

 the one is situated to the right of the centre of gravity, on the 

 same side as the centre of percussion ; the other lies to the left, 

 on the same side as the spontaneous centre. 



If we wish to consider the distance S between the centre of 

 percussion C and the points in question, then, since this distance 

 is evidently x—h, the foregoing expression gives 



8= a//?+K2= Vhi; 



that is to say, the distance from the centre of percussion C to either 

 of the two centres of maximum conversion is the geometric mean 

 between the distance h of the centre of gravity, and the distance 1 

 of the spontaneous centre from the same point C. 



This theorem is precisely similar to that respecting the centres 

 of maximum percussion and reflexion (art. 19). Whence we see 

 that, in all bodies, the centres of conversion are the points which 

 would become centres of reflexion, if the motion of the body were 

 changed so that the centre of percussion became the spontaneous 

 centre, and vice versa. 



53. If, in the expression for i9', we substitute for x the first 

 positive root 



we find 



at a "- . 



^- ^2h{h + B)' 



thus 6' has a negative value, in other words, its sign is unlike 

 that of 6. The first centre of maximim conversion, therefore, is 



