NIPHER EVOLUTION OF THE AM. TROTTING-HORSE. 5II 



When the vakies of — ^ are plotted with the simultaneous val- 

 ues of s, we get a somewhat irregular series of points shown in 

 Fig 2, and represented fairly well by the equation 



J'.= A-\-Bs - - - (i) 



(IT ^ 



The constants A and B can be determined graphically with 

 as great precision as the nature of the data will warrant. 

 The values are found to be 



A = 1. 00 



and the differential equation (i) becomes 



||^=- I. „o +0.0110-^ - - (2) 



This equation being put into the form 



5—90.9 °^^° 



it admits of direct integration as follows, 



/: 



ds _ ^ /clT 



1 loy 



5-90.9 

 So T 



on performing the indicated operations 



/(5— 90.9) =/ (50-90.9)4-0.01 10 ^o-o-oiio'^' 

 where ^o and T^ are simultaneous values at any assumed date. 

 Placing the initial values in a single term, we have 



/(^_9o.^)= c— ^r - - (3) 



or for the primitive equation 



C-BT 



s = 90.9 -|-e - - (4) 



where e is the Naperian base. 



It thus appears that the limiting speed of which the trotting- 

 horse is capable, which he will continually approximate and 

 never reach, is 1:31. This follows from (2) by making -||^ = o, 

 or from (3) and (4) by making T= CO . 



