demonstrated by the method of Indeterminates. 79 



"This way of considering the question, it is presumed, will be 

 deemed free from every objection. The principle upon which it 

 rests depending upon the nature of the variable quantities, and not 

 upon their evanescence, (as it is equally true even for constant quan- 

 tities provided they be of different natures,) it is hoped we have at 

 length hit upon the true and logical method of expounding the doc- 

 trine of Prime and Ultimate Ratios, or of Fluxions, or of the Dif- 

 ferential Calculus, &c." 



"This general principle, which is that of Indeterminate Co- 

 efficients legitimately established and generalized, conducts us by a 

 near route to the Indefinite Differences of functions of one or mork 

 variables. 



" To find the Indefinite Difference of any function of x. 

 "Let Li =fx denote the function. 



" Then d u and d x being the indefinite differences of the function 

 and of X itself, we have 



u + du = f(x + dx) 

 Assume 



f (x 4- d x) = A 4- B d x + C d X 2 + &c. 

 A, B, Sic. being independent of d x or definite quantities involving x 

 and constants; then 



u + du = A-}-Bdx + Cdx2+&;c. 

 and by the fundamental principle we have 



u = A, du = B.dx 

 Hence then this general rule, 



" The INDEFINITE DIFFERENCE of any fuuction of X, f x, is the 

 second term in the development q/" f (x + d x) according to the in* 

 creasing powers q/" d x. 



Ex. Let u = x ". Then it may easily be shown independently of 

 the Binomial Theorem that 



(x + d x) " = x " + n . x '^ - • d X + P d X 2 

 .-. d(x") =n.x"-' dx 

 " To find the indefinite difference of the product of two variables. 

 " Let u = X y. Then 



u4-du = (x-|-dx).(y + dy) =xy + xdy-}- ydx + dxdy 

 .•.du = xdy + ydx -f- dxdy 



and by the principle, or directly from the homogeneity of the quanti- 

 ties, we have 



du=:xdy-{-yd x." 



