M. Poinsot on the Percussion of Bodies. 357 



value 



'v/aV + yS*^*^^ 



A 



where 28' is the diameter parallel to the spontaneous axis OS, 

 and 28 its conjugate diameter. By substitution, therefore, we 

 deduce the simpler expression 



Further, let 6 be the angular velocity with which the body 

 rotates around the spontaneous axis OS, and <^ the inclination 

 of OG to OS or, what is the same, of S to S'; it is evident that 

 ^ . OG . sin = ^ . A sin ^ is the velocity of the centre of gravity G 

 of the body, and consequently that the impulse P which set the: 

 body in motion has the value 



P = ni6 . A sin (/>, 

 m being the mass of the body. 



If, in place of P, in the above formula we substitute this its 

 value, and remember that 8S' sin (f) = x/3, we shall find 



^~ aV + ySy + a^^^- 



The factor Ott in this expression represents the velocity of the 

 point D, whose coordinates are a; and y, and the other factor is 

 a part of the whole mass m of the body expressed by the fraction 



«V+/3y + «2/32-5 



hence we may say that, in the movement of the body, the point 

 in question strikes with the same force as it would do if this 

 fraction of the mass were there concentrated. 



Similarly, if D' were any other point having the coordinates x 

 and 7/', tt' its shortest distance from the spontaneous axis, and 

 Q' the percussion wliich it is capable of producing, we should 

 find 



nr_/j^f 'mc^^____^ 



If wc suppose D' to be the reciprocal of the point D, then, by 

 art. 2, wc shall have, between the coordinates of these two points, 

 the relations 



these values of a' and ij, being substituted in the last expression. 



