M. Poinsot on the Percussion of Bodies. 271 



a point by means of which we impart to the body a maximum 

 rotation in an opposite direction to its original rotation 6. 

 If for X we substitute the second value 



x — h — B, 

 which is negative, we find 



and since ^>h, & has a positive value like 6. This second 

 centre of conversion, therefore, is a point by means of which a 

 maximum rotation, in the same direction as the original rotation 

 6, is given to the body. 



Particular cases of the motion of the body. 



54. When the motion of translation u is zero, we find 



.T = 0and^' = 6', 

 that is to say, the centre of maximum conversion then coincides 

 with the centre of gravity, and the original rotation of the body 

 is not altered. 



55. When the rotatoiy motion 6 is zero, we have 



a;=±Kand^'=+^. 



Corollary I. 

 On the points or centres of a given conversion. 



56. To find the points or centres of a given conversion 0, we 

 must make 



^'=-0; 



when, to determine the distances x of these points, we shall have 

 the quadratic equation 



0a?2-Ma; + K2(^ + 0)=O, 



the roots of which will be real, provided 



M2or«2^-2>4K2(02 4-^0). 



This condition being fulfilled, therefore, there are always two 

 centres corresponding to the given conversion 0. On examining 

 the condition, it will be found equivalent to 



© < y . -rrr- or 



U 2h{S + hy 



that is to say, to the self-evident condition that must not have 

 a greater value than that which corresponds to the maximum of 6'. 

 57. Let = n^, n being any given number; the foregoing 

 equation becomes 



nx'^-ax+{n + l)K'^=0, 



