demonstrated by the method of Indeterminates. 79 
“'This way of considering the question, it is presumed, will be 
deemed free from every objection. e 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 MORE 
variables. 
“ To find the Indefinite Difference of any function of x. 
“Let u =fx denote the function. 
“'Then du and dx being the indefinite differences of the function 
and of x itself, we have 
u+du=f(x+dx) 
Assume 
f(x +dx)=A+Bdx+Cdx?+&c. 
A, B, &c. being independent of dx or definite quantities involving x 
and constants; then 
u+du=A+ Bdx+Cdx?-+ &c. 
and by the ban pelos principle we have 
a=A,du=B.dx 
Hence then this general rule, 
‘“‘ The INDEFINITE DIFFERENCE of any function of x, fx, is the 
second term in the development of £(x + dx) according to the in+ 
creasing powers of dx. 
Let u=x". Then it may easily be shown independently of 
the Binomial Theorem that 
ae x? Rens +n.x®-'dx+Pdx?* 
d(x") =n.x"™'dx 
“To find the snllefinite difference of the product of two variables. 
“Let u = xy. en 
u+du=(x+dx).(y +dy)=xy+xdy+ydx+dxdy 
~du=xdy +ydx+dxd 
and by the principle, or directly from the homogeneity of the quanti- 
ties, we have 
du=xdy-+ydx.” 
