290 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Column IX of Table II gives the time that should correspond to 

 each event, according to formulas (4) or (6). Column X gives the de- 

 viation from the record time. Column XI expresses the deviations 

 in percentage of the record time. The mean percentage deviation 

 between observed and computed record times between the limits of 

 1 kilometer (5 furlongs) and 6.44 kilometers (4 miles) is shown in 

 column XII to be 1.9 per cent. Consequently, we may expect by follow- 

 ing the ascending straight line of Figure 6 to predict the record time 

 of any race between these limits to within 2 per cent on the average. 



The mean percentage deviation between observed and computed 

 record times for the entire series of events, i. e. between I mile and 

 4 miles (0.4 to 6.44 kilometers), is 2.4 per cent. 



Horses Pacing. 



The record data for pacing horse-races in harness on American turf 

 appear in columns I, II, and III of Table III. They are taken from 

 page 260 of " The "World Almanac " for 1906. The entries in the suc- 

 ceeding columns, IV to VIII, are then found in the manner previously 

 described. In Figure 7 the logarithm of the time, or log T in column 

 VII, is plotted against the logarithm of the distance, or logZ in 

 column VI. The logarithm of the mean speed in each event, or log V 

 in column VIII, is also plotted. Taking the ascending time-distance 

 line, it is a straight line ruled through the 2-mile record point, so 

 as fairly to conform with the other record points. It makes an angle 

 with the distance axis of 48° 22' or tan~^ |. Referring to the de- 

 scending speed-distance line, it is a straight line drawn through the 

 2-mile record point, so as fairly to conform with the other record points. 

 It makes an angle with the distance axis of — 7° 7' 30" or tan~^ — |. 

 The two straight lines include between them an angle of approxi- 

 mately 55° 30' 30". 



The straight lines of Figure 7 correspond respectively to the following 

 equations : 



log r= flog Z- 1.5363 (8) 



log V = 1.5363 - \ log L. (9) 



These in turn correspond respectively to the following : 



V = '\ meters per second. (11) 



