M. Poinsot on the Percussion of Bodies. 163 



moment of inertia, that is to say, the sum of the products of all its 

 particles by the squares of their respective distances from the axis 

 G Z under consideration. It will be at once seen that the square, 

 whose side is the line K, is merely the mean of the squares of the 

 distances of all the equal particles of the body from the priiicipal 

 axis of rotation, GZ. For each body it is a given, constant quan- 

 tity, dependent solely upon the figure formed by the different 

 points composing that body. 



This being understood, let h be the distance C G from the 

 centre of gravity G to the centre of percussion C, and let us see 

 to what the whole movement of the body, caused by the appli- 

 cation of this unique impulse P, resolves itself. 



II. On the spontaneous centre of rotation. 



5. The force applied at C may be replaced by another equal, 

 parallel, and like-directed force P', applied at G, and by a couple 

 (P, — P), applied to the arm QG=h, and having the moment 

 Vh. 



The force P' = P, applied at the centre of gi-avity C, imparts 



P 



a common \|elocity v = ^ to all the particles of the body; and 



the couple whose moment is Vh, causes the body to turn around 



PA 

 lis principal axis G Z with an angular velocity ^ = ,, „g *. 



In consequence of this double movement, a point 0, situated 



in the production of the line CG beyond the centre of gravity 



and at the distance OG = fl from the same, would be endued at 



one and the same time with two contrary velocities, the one 



P aVh 



» = — , the other fl^ = ^.„y Consequently, in order to find, upon 



the production of CG, that point of the body with respect to 

 which these two contrary velocities are equal, we have merely to 

 set 



P _ aVh 



M ~MK2' 



from which equation we immediately deduce 



and consequently, for the distance a of tlsc point from the 

 centre of gravity G, the value 



* TMorie nouvclle de la rotation des corps, par M. Poii sot, 1" partie, 

 art. IG. 



M2 



