Centres of Percussion and Oscillation. 237 



As to the centre of oscillation, it is the point O of a body ^ 

 or system of bodies, whose distance from F is equal to the length 

 which a simple pendulum must have in order to perform its os- 

 cillations in the same time. We shall see that this centre is the 

 same as the centre of percussion. 



Indeed, when the question relates to gravity, the force p, re- 

 sulting from the action of gravity, exerted upon each material 

 particle of a body, is equal to the whole mass multiplied by the 

 velocity communicated by gravity in an instant to each particle ; 

 that is, 



P = U X Ir, 



u representing this velocity. Moreover this resultant g passes 

 through the centre of gravity ; and consequently its perpendicu- 

 lar distance from the fixed point JF, or from the axis passing 

 through F, is FH; hence the velocity of rotation i?, of any point 

 JJf, when the body is left to the action of its gravity, is 354. 



so that, for the centre of gravity G, the velocity is 



. 



J m r 2 



Now in order that a simple pendulum, whose length is FO, 

 may make its oscillations in the same time with the body L, it 

 is necessary, L being supposed to be drawn from a vertical po- 

 sition by the same angular quantity, that the velocity impressed 

 by gravity at O (fig. 172), perpendicularly to .FO, should be the 

 same as that of the point O (fig. 171); in other words, that it 

 should be to the velocity of G (fig. 171), as FO is to FG. Now 

 by decomposing the velocity u or OP (fig. 172), communicated 

 by gravity in an instant to a free body, into two others, namely 

 OK in the direction of FO, and 00' perpendicular to FO, we 

 shall have 



u : 00' : ; FO : OZ : : FG : FH; 



whence 



M : OO' ; : FG : FH, 



and consequently 



