170 M. Poinsot on the Percussion of Bodies. 



If we wish to consider the distance X between the point T and 

 the spontaneous centre 0, then since this distance is x + a, the 

 above equation gives 



whence results the following theoi-em : — 



" Tlie distance X of the centre of maximum percussion from 

 the spontaneous centre of rotation of the body is the geome- 

 trical mean between the distances of the centre of gravitj^ and 

 of the ordinary centre of percussion from the same point 0." 



20. We may here observe that this theorem is susceptible of 

 another enunciation. For since MK'^ represents the moment of 

 inertia of the body with respect to the principal axis which passes 

 through its centre of gravity, and since a represents the distance 

 between the point and this centre G, M(K- + «^) will, as we 

 know, be the moment of inertia of the body with respect to the 

 parallel axis 01 passing through the spontaneous centre O*. 

 Hence K'^ + a^ is the mean of the squares of the distances of 

 the several molecules of the body from this axis 01. Consider- 

 ing, therefore, the preceding expression 



we may say that "if the body actually turns around a sponta- 

 neous axis, the distance from the same to the centre of maximum 

 percussion is equal to the side of the square which is a mean 

 between the squares of the distances of the several molecules of 

 the body from the axis in question, or in simpler tei'ms, is equal 

 to the arm-\ of inertia with respect to this axis.^' 



21. The double sign + with which this distance X is affected, 

 indicates that there are two points of the same nature at equal 

 distances to the right and to the left of the spontaneous centre 0. 

 The first T falls between the centre of gravity G and the centre 

 of ordinary percussion C ; the second T' falls on the other side, 

 beyond the point O. 



The percussions to which both correspond are maxima, but 

 opposite in direction. The first T is the centre of a maximum 

 percussion greater than the impulse P, but having the same 

 direction. The second T' also corresponds to a maximum per- 

 cussion ; but this is always less than the impulse P, and opposed 

 to it in direction. 



Thus there is a point T with which the body strikes in front, 

 not only more forcibly than with the centre of percussion C itself, 

 but also more forcibly than with any other point; and at the 

 same time there is a second centre T' with which the body also 



* TMorie nouveUe de la rotation des corjos, fe partie, art. 77. 

 t Ibid. 2^= partie, art. 2. 



