490 PROCEEDINGS OF THE AMERICAN ACADEMY 



ance with which a body moving with a variable velocity is said to 

 have at a given instant a velocity which would carry it thirty-two feet 

 in one second. 



To avoid departing too much from well-established usage, the term 

 differential will be frequently used in this paper instead of rate. 



The rate or differential of x will be denoted by D x, and that of f, x 



by Z> (/,*). 



The rate of the independent variable, or the value of Dx, is re- 

 garded as arbitrary in the same sense that the value of x is arbi- 

 trary. . 



Thus, particular values of these two quantities may constitute the 

 data of a question like the following: What is the value of D (a: 2 ), 

 when x has the value 10 and Dx the value 4? 



To differentiate a function of x is to express D (f, x) in terms of x 

 and D x in such a mariner as to furnish a general formula by which 

 D(f, x) may be computed for any given values of x and of Dx. 



Elementary Propositions. 



The following propositions are immediate deductions from the above 

 method of measuring rates : — 



I. The Differential of x-\-h. 



Since any simultaneous increments of x and of x -j- h must be iden- 

 tical, the increments which would be received by each, if they con- 

 tinued to vary uniformly with fhe rates denoted by Dx and D 

 (x-\-h), are equal. Hence the rates are equal, or 



D{x-\-h)—Dx. 



II. The Differential of x-\-y. 



Since any increment of x-\-y is the sum of the simultaneous incre- 

 ments of x and of y, the same relation exists between the increments 

 which would be received if x and y (and consequently x -\-y) con- 

 tinued to vary uniformly with the rates denoted by Dx, Dy, and 



D(x-\-y). Hence 



D(x + y)=Dx + Dy. 



III. The Differential of mx. 



Since any increment of m x must be m times the corresponding in- 

 crement of x, the same relation must exist between the increments 



