* Rotation of Bodies unconfined. 259 



It may be observed, moreover, that it is necessary, in order 

 that the body .Af may really have the velocity w', that the point 

 T of the body L should also have this same velocity V accord- 

 ing to TZ. Let us now see with what velocity this point must 

 advance according to TZ. 



It will have, in the first place, the velocity v common to all 

 the parts of L. Moreover, if we suppose that the infinitely small 

 arc TT perpendicular to G7 1 , represents the velocity of rotation 

 of the point T, by constructing the parallelogram TCT'B upon 

 the directions TT', TA and TZ, we shall have TC for the veloc- 

 ity of T according to TZ in virtue of its rotation. Now the sim- 

 ilar triangles TT'C, GTZ,give 



GT : GZ : : TT : TC = GZ 



But since -o' is the velocity of rotation of the point Z, we have 

 ?' : TT :: GZ : GT, 



whence 



and consequently 



r'X GT 

 11 = ~~~ 



TC - GZ v v ' xGT 

 - X -- 



therefore the total velocity of the point T belonging to the body 

 L, according to EZ, is v + -sf ; and hence v + v' = u' (3). 



If from the three equations found above, in order to express 

 the conditions of the motion, we deduce the values of w 7 , ?/, and 

 v, we shall obtain 



+ LD 2 ) u 



JVw/mr 2 



LJVD* u 



(JV-f L)/ror* + LD* 



