KENNELLY. — AN APPROXIMATE LAW OF FATIGUE. 279 



time, and average speed of the run, in a series of events, any one, say 

 T, plotted against either of the others, say L, the observations are 

 found to fall approximately upon a straight line, with logarithm paper. 

 In other words, the curves i)lotted on plain rectangular paper from the 

 same records are approximately simple exponential curves, of the type 



y = ^■"• 



Instead of plotting the quantities in the ordinary way upon logarithm 

 paper to produce straight lines, we may perform the equivalent opera- 

 tion of plotting the logarithms of the quantities upon ordinary scaled 

 paper, and produce similar straight lines. That is, we may plot any 

 one of the quantities log L, log 7", and log V against either of the other 

 two. For some purposes the latter method is to be preferred, although 

 it takes more time. Its application is presented in Figure 3, where 

 log T and log V are both plotted as ordinates against log L as 

 abscissas for all of the data of Table I. Columns VI, VII, and VIII 

 in the table contain the common logarithms of the entries in columns 

 IV, III, and V respectively. It is seen in Figure 3 that the speeds fall 

 closely upon the descending straight line, as far as the 20-mile dis- 

 tance, as already seen in Figure 2. Beyond the 20-mile distance, the 

 speeds fall off markedly and are much too low to meet the line. 

 Table I indicates, however, that these long-distance records of 30, 

 50, and 100 miles respectively, were made about 50 years ago, whereas 

 the short-distance records are of recent date. At the dates indicated 

 (1846, 1853, 1857) the short-distance trotting records were by no 

 means so good as they are to-day. It is reasonable to assume that 

 if these deviating long distances were attempted to-day, their records 

 would be materially improved. ^ 



^ Since this paper was written, the writer has been indebted to Prof. E. L. 

 Mark for a photographic curve-sheet pertainino; to a paper presented by Prof. 

 Francis E. >«"ipher to tlie St. Louis meeting in 1903 of the American Association 

 for the Advancement of Science. The curve-sheet shows the steady reduction 

 in the record times of the trotting-horse mile and also of the running-Iiorse mile 

 at diiferent dates between 1840 and 1903. The curves indicate a final limit to the 

 trotting mile at 98 seconds and a final limit to the running mile at 91.5 seconds. 

 The equations to the curves do not appear on the sheet, but have been computed 

 by the writer, from the curves, as follows : — 



At any epoch y years after 1840, the trotting-horse mile record appro.xi- 

 mates to 



Ty = 98 (1 + 0.56 X 10-o"0526y) seconds (a) 



and for the running-horse mile record: 



T, = 91.5 (1 + 0.154 X 10-0009932) seconds (b) 



where z is the epoch in years after a.d. 1863. 



