RHEOMETRY. 3C9 



Resuming the equation y ax -f 5, which is that of a straight 

 line, (p. 335), if we give to x an increment h, and the correspond- 

 ing increment of y be k, then the equation becomes y -{ ka (x -f h) 

 -f b = ax -f ah + 6 , and, from this, subtracting the original equa- 



k 

 tion, we have k = /&, or = a ; which shows that in an equation 



of the first degree, corresponding to a straight line, the increments 

 of the variables, or of the ordinate and abscissa, have a constant ratio. 

 But if we take the equation of a circle, y 3 -f- a? 2 = -K 3 , or y* = R 2 

 X s , which is an equation of the second degree, hence belonging to a 

 curve of the second order, and if we give to x and y the correspond- 

 ing increments h and Ar, we have (y -f- A;) 2 =R* (x + A) a ; or y 2 -f- 

 2 ky -f- & 2 = 7? 3 x z 2 hx A 2 : and subtracting the original 

 equation from this, we have 2 ky + k 3 = 2 hx /i 3 ; in which the 

 ratio of k and h varies, whether we vary x and y, or change the 

 values of k and h themselves. But if we make these increments 

 infinitely small, their squares k z and 7i 3 will become infinitely small, 

 even compared with k and h; and hence may be neglected; so that 



Tf <y 



we then have the equation 2 ky = 2 hx; or = . In this 



case, k and h become the differentials of x and y, and are expressed 

 by writing the letter d before the quantity from which they are 

 derived. 



Thus, when we pass to the limit, by making k and y infinitely 



dij x 



small, the last equation becomes = : from which we have 



x dx y 



dy = dx, for the first differential equation of the circle. The 



y oc 



ratio of the increments, that is, , is technically called the diffe- 

 rential co-efficient ; and it expresses the tangent of the angle which 

 a tangent line to the circle, at the point whose coordinates are x and 

 y, makes with the axis of abscissas. This furnishes us with an 

 easy mode of drawing a tangent line to the circle, at any point 

 whose coordinates x, and y are given. Moreover, if the value of y, 

 after increasing to a certain extent, should reach its greatest limit, this 

 will be shown by dy becoming infinitely small, or zero, in compari- 

 son with dx ; that is, we shall have in this case -~ = ; or = ; 



ax y 



or # = y x = 0, showing that the greatest value of y is that 

 which corresponds to x 0. From this, the maximum value of y 

 itself may be found, by making x = in the original equation ; 

 which, for this value, gives y=R. The minimum value of y, is 

 found on precisely the same principle, and by the same method ; 

 from which we perceive the ready application of this Calculus to 

 problems of maxima and minima. 



We have only room remaining to give some of the simplest rules 

 for differentiating quantities, in order to find the differential equations. 

 The differential of ax, is a . dx ; and that of a constant term is zero ; 

 so that constant terms have no influence on the differential equation ; 

 as shown by the term # 2 in the last example. The differential of 



