264 Dynamics. 



fmr* + L X GZ 

 ~GZ xL ' 



'm r 3 -f L X GZ is in article 360 precisely what we have 

 understood by^'m r 2 in article 361 ; therefore the point Fis here 

 the same as the point O in article 361. 



We perceive, therefore, that the point about which a body 

 may be considered as turning for an instant, is independent of 

 the value of the force or forces which are applied to this body ; 

 and generally it may be inferred from the value of FG, that this 

 point is the more distant, according as the force in question, or the 

 resultant of all the forces, acts at a less distance from the cen- 

 tre of gravity. 



361. 384. We have seen that when a body turns about a fixed 

 point or axis, its centre of percussion is the same as its centre of 

 oscillation ; whence these two centres are found by the same op- 

 eration. It is not the same when the body is free. For, let us 

 suppose a body whose mass is L, to turn about its centre of grav- 

 ity with a velocity, which, for a point situated at the known dis- 

 tance a, shall be v ; and that at the same time this centre moves 

 with the velocity u. It is manifest, in the first place, that the re- 

 sulting force of all the motions belonging to the different parts 

 of this body, will have for its value L x u or L u, that is, the 

 same as if the body had no motion of rotation. In the second 

 place, the distance at which the resultant must pass from the 

 centre of gravity, is evidently that at which a force equal to L w, 

 would produce in the body a velocity of rotation equal to that 

 which it actually has ; but this velocity has for its expression 



> calling D the distance sought ; we have, therefore, 

 fmr 2 



_ LuDa 

 " ~fmr 2 '' 



and consequently 



