M. Poinsot on the Percussion of Bodies. 17D 



must be presented in order that, after the shock, the body may 

 turn around its own centre of gravity in a direction eitlier con- 

 trary to, or the same as its present rotation 0, and that with the 

 greatest possible or, if we please, with a given angular velocity. 



Here, tlien, are new and remarkable points in the body pre- 

 sented for consideration, points which it will be convenient to 

 distinguish by particular names. 



The first may be called centres of reflexion or j^rogression 

 according as the centre of gravity of the body is reflected, or 

 continues its forward motion with the new velocity in question. 

 The term centres of reflexion, however, may be applied to both, 

 provided we admit that this reflexion may be positive or nega- 

 tive; positive when the body actually rebounds, and negative 

 when it pursues its course with the new velocity it has acquired. 



In a similar manner the second points may be called centres 

 of conversion ; this conversion being regarded as negative when 

 the body, instead of changing the direction of its actual rotation 

 6, continues to turn on itself in the same direction as before. 



On the centre of ^^aximum reflexion. 



33. In order, then, to find the point against which, if the 



body strikes, it will be reflected with the greatest possible velo- 



du' * 



city, we have merely to make «'= a maximum, or'-j-=0. By 



so doing, the distance x of this centre will be found from the 

 quadratic equation 



ex'^ + 2ux-eY^^-0; 

 whence, replacing u by ad, we deduce 



x=z—a± Va'^+K^=-a±\, 



an expression identical with that found in art. 19 for the distance 

 of the centre of maximum percussion. 



Thus the centre of greatest reflexion is the same as that of 

 greatest percussion ; a result which might have been immediately 



du' 

 seen, smce the equation -— =0, which gives the value of x cor- 

 responding to the maximum of u', is the same as the equation 

 — =0; but the latter is itself the same as ^ =0, which cor- 

 responds to the maximum of Q. 



34. Of the above two values of x, the first, \ — a, which Is 

 positive and less then h, gives for u' the value 



i/=_ 



{\-a)e 



2 ' 



which is negative, or in other words, has a sign unlike that of 



N2 



