Rotation of Bodies unconfined. 263 



line GZ is carried parallel to itself from GZ to G'Z', we imagine 

 it to turn about the moveable point G, since the points of the 

 body have velocities of rotation greater in proportion to their 

 distances from G, it is manifest that there is upon the line ZG a 

 point F which will be found to have described from F' toward 

 I 1 , an arc equal to GG', and which may be regarded for an in- 

 stant as a straight line ; the point F then will have retrograded 

 as far by its motion of rotation as it has advanced by the \elo_c- 

 ity common to all parts of the body ; this point will therefore 

 have remained constantly in F, which, for this reason, may be 

 considered for an instant, as a fixed point about which the body 

 turns. If we would know the position of the point F, it will be 

 remarked that the arcs FF', Z'/, which the points F' and Z' de- 

 scribe in an instant, may be considered as straight lines perpen- 

 dicular to GZ, or parallel to GG' ; now the similar triangles 

 FF'G', G'Z'/, give 



G'Z' : G*P : : Z'l : FP, 

 pr 



GZ : GF :: Z'/ : GG'; 



but we have found the velocity, 



GG' = --, and the velocity Z'/ = ~^- 5 



hence 



GZ or D : GF : 



f m r 2 L 

 therefore 



GF= /^l. 



383. The point F is called the centre of spontaneous rotation, 

 because it is a centre which the body takes as it were of itself. 

 This point is precisely the centre of oscillation which the body 

 L would have, if it turned about a fixed point or axis situated in 

 Z ; for from 



GF = 



we have 



