CH. X TIME CUKVES 219 



value of E, the final speed will vary directly as the first 

 power of /3; We may express this as fpllows : 



(99), 



where A' 2 is a constant. 



It appears then that by varying the diameter of the 

 driving-wheel, the induction factor and the velocity ratio, 

 we can vary the way in which the given distance is covered. 

 If, for instance, we put on a small wheel, we shall get a 

 high acceleration and a low final speed, which will soon 

 be reached, and most of the distance will be covered at 

 full speed. If, on the other hand, we use a large wheel, 

 we shall get a low acceleration and a high final speed, 

 which will be reached only after a considerable lapse of 

 time, perhaps not at all, and most or all of the distance 

 will be covered during the accelerating period. Between 

 these extremes there is a certain value of J3 that will 

 enable us to cover the given distance in the shortest time ; 

 this we now proceed to determine. 



The area hafg may be expressed thus : 



brj ............ (100). 



A; D Jc 



We have also &/i= -^ /3 2 and bq= ,_ _. i^/3 2 ; hence 

 fc, Jc 2 S 2 \ 



the time occupied is given by 



+(3* ........ (101). 



TO find the value of /3 that makes this a minimum, 

 differentiate and equate to nothing ; this gives us 



