264 M. Poinsot on the Percussion of Bodies. 



by means of which the unknown distances of these centres are 

 determined. The two roots of this equation will be real, provided 



and, as may be easily seen, both roots are positive, and less than 



h or — . 

 a 



Hence, provided V satisfies the inequality 



there are two centres in the body corresponding to the given 

 reflexion V. On examining this inequality we find it to be 

 equivalent to 



{\-a)d 



and consequently that it expresses the self-evident condition that 

 the given value V of u' must not exceed the maximum of u'. 



38. If we represent V by nw, n being any number whatever, 

 and subsequently replace u by ad, the foregoing inequality 

 becomes 



K 



2 Vn^ + n 



hence the theorem : — If the movement of the body be such that the 

 distance a between its centre of gravity and the spontaneous centre 



of rotation is less than ^ — , , then between the centres of 



'' ^Vn^ + n 



gravity and percussion there will always be two points or centres, 

 such that if an obstacle be presented to the one or the other, the 

 body will be reflected with a velocity V, n times greater than that 

 which it possesses. 



39. If, as a particular case, 



K2^2=4(V2 + Vh), 

 tlie two roots of the preceding equation will be equal, and the 

 two centres in question will coincide with the point whose di- 

 stance from the centre of gravity is 



But, solving the corresponding equation 



4,V2 + 4Vm=K2^ 

 for V, we find 



{\-a)d 

 2 ' 

 that is to say, the given value of V is in this case equal to the 



