236 Dynamics. 



2 



FF' x L, the mass of the body being represented by L. For 

 the same reason/ w x ZFP x F'P is simply 2 FF'fm X PP. 

 But/ m X F'P, being the sum of the products of the particles 

 into their respective distances from a plane passing through A'B', 

 64 that is, through the centre of gravity, must be equal to zero ; we 

 have therefore simply, 



fm'mF=fm'mF' + Lx FF'. 



Hence, knowing the exponent f m m F' of the moment of inertia 

 with respect to an axis passing through the centre of gravity, we have 

 the exponent with respect to any other axis parallel to this, by adding 

 to the first the product of the mass into the square of the distance 

 between the two axes. 



354. From this result, and the expression for the velocity of rota- 

 tion, it may be inferred that of all the axes about which a body may 

 be made to turn in virtue of any force or impulse, those about which 

 the velocity of rotation will be the greatest are such as pass through 

 the centre of gravity ; since the exponent of the moment of inertia 

 with respect to an axis passing through the centre of gravity, is 

 less than it is with respect to any other axis. 



Of the Centre of Percussion and the Centre of Oscillation. 



361. The foregoing propositions will be found to be of the 

 greatest importance in many inquiries to be resumed hereafter ; 

 we shall confine ourselves for the present to the use that may be 

 made of them in finding the centre of percussion, and centre of 

 F\" m. oscillation, of bodies that admit only of a rotation about a deter- 

 minate axis or point. We understand by the centre of percus- 

 sion, the point. O of the straight line FG, where it would be nec- 

 essary to place a body in order that it might receive the greatest 

 impression from the body L turning about F. Now it is evident 

 that this point must be that through which passes the resultant 

 of the motions of rotation of all the particles in L. The centre of 

 percussion, therefore, is determined by the proposition of arti- 

 cle 358. 



