Maximum Effect of Agents. 275 



greater velocity ; but there is an obvious limit to this, for he 

 would soon be fatigued by the rapid walking, and would 

 therefore be rendered unfit to continue his work. He must 

 therefore return to that distance from the axis, where the 

 wheel has such a velocity that he can continue to move with that 

 velocity during the period that his work is to last; that is, 

 there is a particular velocity with which the man must walk, in 

 order to perform the greatest quantity of work ; and it would 

 be easy to find this velocity, if we knew the law according to 

 which his force is diminished, as his velocity increases. We may 

 suppose, however, that his force diminishes in the same ratio as 

 his velocity increases. 



393. Let/? represent the force which a man can exert during 

 a given time against a dead weight. This force will .obviously 

 be greater than any which he would exert on the supposition of 

 motion taking place ; for a part of his strength in this case would 

 be expended in putting himself in motion and in continuing this 

 motion. Let v be the velocity with which he would lose the 

 power of exerting any force ; then, if he move with a velocity v* 

 less than , he will exert a force less than jo, and the part lost 

 may be found, according to the above hypothesis, that the dimi- 

 nution of force is as the increase of velocity. Since he loses all 

 his force, or /?, when the velocity is z>, and none when there is no 

 velocity, we have 



v or v : i/ or &' : : p : ^ -. 



-j 



- is therefore the loss of force sustained on account of moving 

 with the velocity v'. There will accordingly remain 



as the effective force actually exerted against the weight. Now 

 if D be the distance at which this force acts, r the resistance or 

 weight raised, and tf the distance at which the resistance acts, 

 and u its velocity ; then, when the machine has attained a uniform 

 motion, we shall have 



