274 MOTION UNDER A VARIABLE FORCE 



Moment of a Velocity 



219. The velocity of a point is a vector, and the line of action 

 of this vector may be supposed to be the line through the moving 

 point in the direction of its velocity. We can define the moment 

 of a velocity in just the same way as the moment of a force has 

 been defined. Moreover, all the properties of the moments of a 

 force followed from the fact that forces could be compounded 



according to the parallelogram law, so 

 that the same properties will be true of 

 the moments of velocities, because veloci- 

 ties also can be compounded according to 

 the parallelogram law. 



Let P be a particle describing an orbit 

 about 0, and let OQ be a perpendicular 

 from on to the line through P in the 



direction of the particle's velocity. Then the moment of the par- 

 ticle's velocity about is OQ x (velocity of particle). 



After a short interval dt, let the particle be at P'. Its velocity 

 at P r is compounded of its velocity at P together with dt times its 

 acceleration at P. Hence 



(moment about of velocity at P r ) 



= (moment about of velocity at P) 



4- (moment about of [dt x acceleration at P] ). 



The acceleration at P being along PO, the last term of tl 

 equation is zero, so that we see that the moments about of tl 

 velocities at P and at P' are equal. 



We can extend this step by step as in the former theorem, 

 obtain finally that 



The moment about of the velocity of a particle describing 

 orbit about is constant. 



220. We have supposed that the particle moves from P to 



in time dt, so that if v is its velocity at P, then PP' = v dt. As 

 the particle describes its orbit, the line OP may be regarded as 



