84 KINEMATICS. [162. 



162. The meaning of these expressions will perhaps be better 

 understood by the following geometrical derivation. As shown 

 in Art. 142, the velocity v has the components 



dr <te 



the former along the radius vector, the latter at right angles 

 to it. During the element of time dt, while the moving point 

 passes from P to P' (Fig. 40), each of the vectors v r , V Q 



Fig. 40. 



receives a geometrical increment V r V' n V e V l V Let us resolve 

 each of these infinitesimal vectors along r and at right angles 

 to r, and the'n combine the two components along r, and also 

 the two components perpendicular to r\ finally, dividing by dt, 

 we obtain/. and/ fl . 



Thus v r gives along r y and at right angles to r y 

 dt* dt dt 



hile z/0, or r-, contributes r I ) along r and 

 dt \dt ) 



w 



^^^ = 4- 

 dt\ dt) dt dt dt* 



at right angles to r. We obtain in this way the same expres- 

 sions for/,.,/^ as in the formulae (6) above. 



