28 JOURNAL OF THE 



du 



dii dt rate of change of // 



dx dx rate of change of x 



dt 



As an illustration, find the rate at which the volume u 

 of a cube tends to increase in relation to the increase of an 

 edge X, due to a supposed continuous expansion from heat. 



du 

 u = x^ .' . — ^ 3.i'l 

 dx 

 Therefore for x = i, 2, 3, the volume tends to increase 

 at a rate 3, 12, 27 times as fast as the edge increases. 

 Numerous examples could be given of the application of 

 the differential calculus to the ascertaining of relative rates, 

 but the above will suffice to illustrate the principle. 



dii 

 It has been shown above that if // =zf[t)^ — represents 



dl 



the rate of change of ?/, if at any time /, the rate is sup- 

 posed to become uniform; hence die represents what the 

 increment of 2c would become in time di. 



As time must vary uniformly, dt is always a constant^ 

 though it is entirely arbitrary as to numerical value; hence 

 the differential of a variable can be defined^ as zvhat the 

 increment of the variable would become in any interval of 

 time if at the instant considered^ the change becomes uni- 

 form or the rate becomes constant. If 21 is a function of 

 several variables, then the differentials of each must be 

 simultaneous ones, corresponding to the same interval of 

 time. 



Newton, in establishing his calculus of fluents and flux- 

 ions, conceived a curve to be traced by the motion of a 

 point, an area between the axis of .f, the curve and two 

 extreme ordinates, to be traced by the motion of a variable 



