OF ARTS AND SCIENCES: JANUARY 14, 1873. 491 



which would be received if x (and consequently mx) continued to 



vary uniformly with the rates denoted by D (x) and D(mx). 



Therefore 



D (/» x) =m D x. 



The Ratio of the Rates of a Variable and its Function. 



Let y denote a linear function of x such that 



y=zmx-\-b. [1] 



By propositions I. and III. 



D y = m D x, 



P x = m. [2] 



In this case, the ratio of the rate, or differential, of the function to 

 that of the independent variable is constant, its value being indepen- 

 dent not only of x, but also of Dx. Thus, if we give to Dx any 

 arbitrary value, it is evident from equation [2], that Dy must take a 

 corresponding value such that the ratio of these quantities shall always 

 retain the constant value m. 



Assuming rectangular co-ordinate axes, if y be made the ordinate 

 corresponding to x as an abscissa, the point (x,y) will, as a; varies, 



generate a straight line. The direction of the motion of the point is 



D y 

 constant, and depends upon the value of m. Since =r- is equal to 



m, it is the trigonometrical tangent of the constant inclination of the 

 direction of the generating point to the axis of x. 



When y is not a linear function of x, the direction of the motion of 



the generating point is variable, and consequently the value of =— is 



variable. 



Making, now, the arbitrary quantity Dx a constant, Dy will be a 

 variable. Suppose, then, that, the generatrix having arrived at a given 



point, the ordinate y continues to vary uniformly with the rate de- 



Dy 

 noted by D y at the given point; the value of ^r— will become con- 

 stant. The generatrix will now continue to move uniformly in the 



direction of the curve at the given point, and therefore the value 



D y 

 which yt nas at ^ s P omt i 3 tnat °f tue trigonometrical tangent of 



the inclination of the curve to the axis of x at this point. The line 



