THE GEOMETRY OF MOTION 229 



the length P^Pg is constant and equal to IV^ ; hence if 

 the point P traverse this length in a number of minutes, 

 which we will represent by the letter t^ we must have, 

 since speed is the number of units of length per minute, 

 the length P^P^ equal to the product of IV^ and t (or in 

 symbols P^P^ = IV^ x f). Further, since the angle L^NL. 

 is turned through by the tangent also in time /, the ratio 

 of the angle L,NL^ to / is the mean rate at which the 

 tangent is turning round in the time /, or is the mean 

 spin of the tangent (or, if the mean spin be denoted by 

 the letter S, we have in symbols Z. L^NL^ = S X /■). From 

 these results it follows at once that the mean curvature 

 which is the ratio of L^NL^ to P^P^ must be equally the 

 ratio of the mean spin S to the mean speed IV^. Thus 

 we have directly connected motion with curvature. 



Proceeding in conception to the limit we have the 

 important kinematic result that : If a point moves along 

 a curve the ratio of the spin of the tangent to the speed of 

 the point is the actual curvature at each situation of the 

 point. 



It remains to connect this result with the acceleration. 

 The acceleration in the case we are dealing with is the 

 velocity of V along its circle (Fig. 15). This acceleration 

 at V^, for example, is along the tangent V^T^ to the circle, 

 or at right-angles to IV^ the direction of the velocity of 

 P (Fig. 14) ; it has thus, as we have seen, purely a shunt- 

 ing and no spurting effect. Now, since IV^ and IV^ were 

 drawn parallel to the directions of motion L^P^, L^Pg at 

 P^ and P5 respectively, it follows that the angles L^NL^ 

 and V^IVj — between two pairs of parallel lines — must be 

 equal. Hence the mean spin of the tangent from P^ to 

 P. must be the ratio of the angle V,IV. to the time t in 

 which P passes from P^ to P,, or, what is the same thing, 

 in which V passes from V^ to V,. But the magnitude of 

 the angle V^IV^ is (see the footnote, p. 227) the ratio of 

 the arc V^V^ to the radius IV^. Further, the ratio of the 

 arc V^V. to the time / is the mean speed of V from V^ to 

 Vg (p. 218). Thus it follows that the mean spin of the 

 tangent (Fig. 14) is the ratio of the mean speed of V to 



