258 Dynamics. 



ner as to cause no rotation in L, except about a single axis per- 

 pendicular to the plane which passes through the centre of 

 gravity G, and the perpendicular TZ belonging to the point of 

 contact T\ it is proposed to determine the velocities after collis- 

 ion, and the directions of these velocities, the body L being 

 supposed at rest. 



Let us imagine a plane touching the point T, and let the ve- 

 locity of JV, according to EQ be decomposed into two others 

 one according to ET perpendicular to this plane, and the other 

 according to El parallel to this same plane. If JV had no other 

 velocity but 7, it would only touch L in passing, and would 

 communicate to it no motion, the effect of friction being out of 

 the question. It is therefore only in virtue of the velocity ET^ 

 that the impulse is produced. Now as it is easy to determine 

 ET in the parallelogram 7^7, of which all the angles and the 

 diagonal EJi are supposed to be known, we shall consider this 

 velocity E T as known, and we shall call it u. Let u' represent 

 the velocity of .AT after collision, according to the direction ET 

 or CZ ; consequently u u' is the velocity lost by collision, and 

 N X (u u f ) is the force impressed upon the body L, which 

 we have called p. Therefore the centre of gravity and all the 

 parts of the body will move in the direction GM parallel to CZ, 

 with a velocity 



* = ^F^ (o, 



calling this velocity v. 



But, as the force JV X (u u') does not pass through G, the 



centre of gravity of L, the body must turn about G, as if this 



136 point were fixed. Let v' be the velocity of rotation of the point 



Z where GZ, perpendicular to CZ, meets the latter line ; we shall 



have, therefore, 



T/ - -^ * ( x uz 

 or representing GZ by Z>, 



T)' = ^^ ^ ~~ M/ ) 



j mr* (2). 



