273 M. Poinsot on the Percussion of Bodies. 

 whence 



2n 

 and both roots are real and positive, provided 



«2_4K2(tt2 + „)>0. 



Hence when the' spontaneous centre of rotation is sufficiently distant 

 from the centre of gravity to make 



a > 2K Vn^ + n, 



there are always two points of the body, on the same side of its 

 centre of gravity as the centre of percussion, such that if either 

 becomes the point of impact the body will suddenly commence rota- 

 ting in an opposite direction loith a velocity n times greater than 

 that which it possessed before the shock. 

 When fl = 2K\/n^ + w, both these centres coincide with one 



and the same point situated at the distance 5- from the centre 



of gravity. 



58. If we now suppose n to be negative, it is evident that 

 when the inequality 



a>2KVn^~n 



is satisfied, there are tivo points such that, if the body strikes an 



obstacle ivith either, it loill also commence a rotation n times greater 



than it had before the shock, but in the same direction as the latter. 



If, further, a = 2K \/n- — n, these two centres will coincide 



with a point at the distance — g— from the centre of gravity. 



Corollary II. 

 Centres of perfect conversion. 



59. To find the points by means of which the body might be 

 made to assume a rotation equal and opposite to tliat which it 

 possesses, we have merely to make 



^'=-^orn = l 

 in the expressions of art. 57. In this manner we find 

 a+ V-«2-8K2 

 2 

 both of which values are real and positive, provided 

 a2-8K2>0. 



60. If we were to make 



^' = ^or n=-l, 

 we should find ^2^^^,_o 



