NIPHER — EVOLUTION OF THE AM. TROTTING HORSE. 515 



tervals dT must be increased in order to bring Mr. Pickering's 

 values of -^1^ iiito accordance with the values calculated from 

 the above equation. In this case the intervals are supposed to be 

 separately adjusted. If the later dates were simultaneously ad- 

 justed or changed by intervals ranging from two-tenths to three- 

 fourths of a year, the values of —^ , which Mr. Pickering pre- 

 fers to use, would agree exactly with the values calculated from 

 the last equation. Now it is perfectly clear that these later dates, 

 and particularly the last two, are subject to just such errors as this. 



Whatever these values of -^ may be said to prove, they cer- 

 tainly do not prove that my results as before published are absurd, 

 and they do not indicate a limiting speed of _ 25 seconds. If 

 -^ = o, the limiting speed of the horse is found to be 98 seconds. 



I desire to express my thanks to Mr. Pickering for his criti- 

 cisms and suggestions, as he has corrected a tendency which I 

 had begun to feel, to attach too much importance to the numeri 

 cal result reached ; but I maintain that his method, correctly ap- 

 plied, gives in general, substantially, the same result as my own. 

 It is not necessary to assert that this result is really correct if any 

 person feels inclined to doubt it, because at present it is not pos- 

 sible to demonstrate it more fully than has been done in the pres- 

 ent paper. I only insist that it is not wholly unwarranted by the 

 facts which we now know. 



Most horsemen seem to think that the limiting speed of the 

 trotting-horse will be somewhere near a mile in 120 seconds. If 

 this were true, the differential equation could hardly be a linear 

 one. The equation 



where i- is a constant and L is the limiting speed, would how- 

 ever be in harmony with this view. But this equation gives on 

 integration an equation of the form 



ojJ^^L = C—A T - - - - (8) 



According to this equation the horse would absolutely reach 

 the limiting speed Z in a finite time, -?. Practically this may be 

 true, as is in fact shown by my own equation (4J, so that some 

 such equation might really represent the results sufficiently near 



