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PROCEEDINGS OF THE AMERICAN ACADEMY. 



ratio of all is 2.173, which is within 0.4 per cent of the value 2.181 

 required in equation (28). This result constitutes an independent 

 demonstration of the approximate accuracy of that equation. If pairs 

 at short distances affected by starting retardation, or at very long 

 distances, had been rejected, the agreement would have been still 

 closer. 



Equation (28) leads to the following : 



T = -TT^ seconds 

 y " 



(29) 



or the time varies approximately inversely as the ninth power of the 

 speed in the race. If we assume that the racer reaches the winning 

 post virtually exhausted, so far as affects racing eff"ort, then the time 

 of exhaustion varies inversely as the ninth power of the speed. The 

 computed eff'ect of increasing speed is given in the following table : 



TABLE XIV. 

 Computed Infltjence of Speed upon the Time of Exhaustion. 



Table XIV shows that if a racer increases his speed 10 per cent, he 

 brings down his running time from 100 to 42.4, or 2.36 times ; and if 

 he doubles his speed, he becomes exhausted in 0.195 per cent of the 

 original time, or 512 times more quickly. 



The records do not show that any one racer would be exhausted 

 512 times as soon if he doubled his speed, because there is no evi- 

 dence at hand as to the behavior of any single individual at doubled 

 speeds. If, howevei-, any one racer could be trained to take the record 

 speed at each event in the entire series, — that is to say, if a runner 

 could be trained to take every race from the 20-yard dash to the 50- 

 mile run at world's record speeds in each, — then this ideal athlete 

 would be exhausted in a time approximately as the inverse ninth 

 power of the speed. It is reasonable to assume that what would be 



