326 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



It is thus indicated that with 500 times more time, the distance 

 covered will be only 251 times greater. As an example, we may take 

 the 4-mile (6.44 kilometers) walking event of table V. It occupied 

 1658 seconds. If we increase the time twenty times, or to 33,160 

 seconds, we should expect from Table XV that the distance covered 

 would be 57.2 miles (92 kilometers). The nearest event to this in 

 Table V is the 60-miles (96.6 kilometers), occupying 34,847 seconds or 

 21 times the original time, which is a satisfactory agreement. 



Another consequence of equation (28) is expressed : 





L '^^ ^ meters 



(31) 



or the distance covered in a race varies approximately as the inverse 

 eighth power of the speed adopted. That is, if an athlete could be 

 trained to take any distance from the shortest to the longest at the 

 record speed for the event ; then the distance which this athlete would 

 be able to run before being exhausted would be as the inverse eighth 

 power of his speed over the course. A few numerical values are given 

 in the accompanying table : 



TABLE XVI. 

 Distances capable of being traversed as the Speed is increased. 



The table shows that if the speed is doubled, the distance that can 

 be run, before exhaustion, is reduced 256 times according to (31). 



As an instance, the 20-mile (32.19 kilometers) running event of 

 Table IV, already referred to, may be selected. The speed over the 

 course was 4.794 meters per second. At 9.588 meters per second the 

 distance should be -^-^ = 0.0781 mile = 125.7 meters. Table IV shows 

 that at 9.696 meters per second, the nearest to the required speed, 

 the distance run was 131.5 yards, or 120.2 meters. 



