266 M. Poinsot on the Percussion of Bodies. 



should have the equation 



or putting a6 and /* in place of u and — , respectively, the sim- 

 pler equation h K^ ^ 



whence we deduce the roots 



h , -v/A2_8K2 



0!= -r ± , 



4 4 



which are always real, positive, and less than h, when 



/i2-8K2>0, 



or, putting — in place of h, when 



K 



It follows, therefore, that if the movement of the body is such that 

 the distance a, from its spontaneous centre to its centre of gravity 



G, is less than the line -k—j^, there are always two points or 



centres of perfect reflexion ; that is to say, two points such that if 

 the body strike an obstacle with either, its centre of gravity will be 

 reflected with a velocity exactly equal to that which it possesses. 



Examples. 

 45. Suppose, by way of example, thata=^K: the condition 



K 



-^ is evidently fulfilled, and we have for x the two posi- 



2\/2 

 tive values 



Xj='K and x,i=^'K.. 



The two points on the line CG, joining the centres of gravity 

 and percussion, which correspond to these distances, are there- 

 fore centres of perfect reflexion; so that by presenting an obstacle 

 to either of them, the centre of gravity of the body would be 

 reflected as if the latter were perfectly clastic. 



Although the centre of gravity will move in the same direction 

 and with the same velocity, no matter whether the body strikes 

 with the first or with the second of these points, it will easily be 

 seen that the rotation 6' which the body preserves after the shock 

 is not the same in both eases. In fact, substituting, successively, 

 these values of a- in the expression for 6' (art. 31), we find that 



9 20 



the rotation 6' becomes ^ in the first case, and -^in the second. 



