280 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The ascending line in Figure 3, connecting log T and log L, repre- 

 sents the sequence of record times with satisfactory precision, as far as 

 20 miles, with the exception, perhaps, of the 4-mile event, in which 

 the time is long and the speed low. As shown in Figure 1 or Table 

 I, the speed in the 4-mile trot is only half of one per cent greater 

 than the speed in the 5-mile trot. It should be relatively faster by 

 more than this amount, to judge by the speeds in the other events, and 

 this means that it should lie nearer the straight line of time-distance 

 in Figure 3. 



Beyond 20 miles, the points deviate markedly from the rising 

 straight line of Figure 3 in the direction of excessive time, or low 

 speed, as already considered. 



The rising straight line of Figure 3 represents the equation 



log r=|logZ- 1.53 (1) 



■while the falling straight line of speed-distance corresponds to 



log F= 1.53- i log Z. (2> 



That is, the rising line makes with the axis of abscissas an angle of 

 48° 22', whose tangent is % ; while the falling line makes with the same 

 axis an angle of —7° 7' 30", whose tangent is —I. 



Equation (2) implies that at Z = 1, or upon a course one meter 

 long, the speed of trotting would be 33.9 meters per second, the loga- 

 rithm of this number being 1.53. It would be impossible for a horse to 

 reach any such speed on such a very short course, even with flying 

 start, if only owing to inertia and the large effort required for initial 

 acceleration. The initial velocity of equation (2) is therefore a ficti- 

 tious quantity of merely theoretical interest. The speed curve of 

 Figure 1 and the straight speed lines of Figures 2 and 3 mark a satis- 

 factory application of equation (2) between the limits of 1 mile and 

 20 miles. Columns IX and X of Table I give the computed time for 

 each event, as determined by equation (1) or the rising lines in Figures 

 1, 2, and 3. Column XI gives the deviation or discrepancy between the 



From formula (a) or from tlie curve-slieet, tlie ratios of reduction in tiie 

 trotting-horse mile record time to 1905 are, 



at 1846 0.82 



" 1853 0.844 



" 1857 0.858 



Usint; these correcting ratios, the .30-milc trotting record is brought on to the 

 logaritlnnic straight line in Figure ;'> ; while the 50-niile and 100-inile records are 

 only brought about half-way towards tliat line, as indicateil on the figure. 



