THE GEOMETRY OF MOTION 223 



that the spurting and shunting of motion are conceptions 

 as important for describing our everyday experience as 

 those of the speed and direction of motion itself 



We have seen that the speed changes from the length 

 IV to the length IV, in a certain time — that represented 

 by the length t^^ of our time-chart (Fig. 10). The 

 increase of speed per unit of time (or the ratio of the 

 difference of IV, and IV, to /,/J is termed the mean 



5 4 4 5' 



Speed-acceleration or the mean spiirt between P^ and P^. 

 Further, the ray IV has been turned from IV^ to IV^, or 

 through the angle V^IV^ in time t^,^. This increase of 

 angle per unit time (or the ratio of the angle V^IV^ to 

 / / ) is termed the mean shunt, or mean spin of direction 



4 5' 



between the positions P^ and P^. The two combined, or 



the mean rate of spurting and shunting, form what is 



termed the mean acceleration during the given change of 



position, or for the given time {t^^. What we measure, 



therefore, in acceleration is the rate at which spurting and 



shunting take place. Turning to Fig. i 3 the reader must 



notice that there are two processes by aid of which we 



can conceive the velocity IV^ converted into IV^. In the 



first process we follow the method just discussed : we 



stretch IV^ till it is as long as IV^, that is, we increase 



the speed from its value in the position P^ to its value in 



the position P^ ; then we spin this stretched length round 



I till it takes up the position IV^. This is the spurt and 



shunt conception of acceleration. In the second process 



we say add the step V^V^ to the step IV^ and we shall 



reach the step IV^ (pp. 210-21 i) — that is to say, we can 



consider the new velocity IV^ obtained from the old 



velocity IV^ by adding the step or velocity V^V^ by the 



parallelogram law. The mean acceleration is in this case 



expressed by the step V^V^ added in the given interval 



t^^. But if we compare Figs. 9 and i 3 as maps for the 



motions of P and V we shall see that adding V^V^ in 



time t^t^ corresponds to adding P^P^ in time tj,^. The 



latter operation, however, led us, by aid of the time-chart, 



from the idea of mean speed or mean change in OP to 



the idea of actual speed or instantaneous change in OP at 



