M. Poinaot on the Percussion of Bodies, 265 



maximum of u', Heuce the value K-?i? \ of ,v oueht now to 



2(V + MJ ° 



resolve itself into the distance \—a of the centre oi maximum 

 reflexion. It is easy to verify that it does so. 

 40 Tf 



K2^2^4(V2h-Vm), 



both roots will be imaginary, and no centre corresponding to 

 the given reflexion V can be found. 



41. We may remark that when ^=0, this condition for ima- 

 ginary roots is always fulfilledj whatever value V may have : 

 hence, when the body is simply translated through space, there is 

 no point by means of which it can be reflected: wherever the 

 obstacle may be presented, the centre of gravity, after the shock, 

 will continue to move in the same direction as before, and that 

 always with diminished velocity. 



42. When u — 0, that is to say, when the body merely pos- 

 sesses a rotation around its own centre of gravity, there are 

 always two centres corresponding to a given reflexion V, provided 



V is less than -^r-. This accords with the result in art. 36, since 



-^ is the maximum velocity that can be given to the centre of 



gravity of a rotating body by presenting to the latter an obstacle 

 or fixed point so as to oppooc its rotation. 



43. If we replace V by —V throughout the preceding ana- 

 lysis, we shall have the case of a negative reflexion, in other 

 words, of a ^progression of the centre of gravity in the same direc- 

 tion as before. Hence to determine these centres of a given 

 progression, or these points by means of which the body, after 

 striking an obstacle, continues to move in the same direction as 

 before with a given velocity, we shall have the equation 



(m - V)a;2 _ K2^^ _ VK2 r= 0, 



the roots of which are real, equal, or imaginary, according as 



K^^-4(V2-Vm) >0, =0, or <0, &c. 

 The discussion of this equation would be analogous to the fore- 

 going. 



Corollary. 



On the centres of perfect reflexion. 



44. If we wished to consider in particular those points or 

 centres by means of which the body would be reflected with the 

 same velocity as it possesses, and thus deport itself as a perfectly 

 clastic body, we should merely have to make 



m'= — m, 



or to set Y=u in the formulae of art. 37. By so doing we 



