76 Fundamental Principle of the Higher Calculus 
“The difference between a Definite value and the Indefinite value 
of any quantity y is Indefinite, and we call it the Indefinite Difference 
of y, and denote it, agreeably to the received algorithm, by 
dy | 
‘In the same manner 
d (dy) 
or 
d?y 
the Indefinite Difference of the Indefinite Difference of y, or the 
second indefinite difference of y. 
“Proceeding thus we arrive at 
which means the n™ indefinite difference of y.” 
“The reader will henceforth know the distinction between Defi- 
nite and Indefinite Differences. We now proceed to establish, of 
Indefinite Differences, the 
FUNDAMENTAL PRINCIPLE. 
“It is evidently a truth perfectly axiomatic, that no aggregate of 
INDEFINITE quuntitres can be a definite quantity, or aggregate of 
definite quantities, unless these aggregates are equal to zero. 
“It may be said that (a—x) + (a+ x) =2a, in which (x) is in- 
definite, and (a) constant or definite, is an instance to the contrary ; 
but then the reply is, a—x and a+ x are not indefinites in the sense 
of our definition. . : 
“* Hence if in any equation 
A+ Bx+Cx?+Dx2+4&.=0 
A, B,C, &c. be definite quantities and x an indefinite quantity ; 
then we have 
A= 0, B=0,C = 0, &. 
“For Bx + Cx?4+Dx*+ &c. cannot equal — A unless A = 0. 
But by transposing A to the other side of the equation, it does = — A. 
Therefore A = 0 and consequently 
Bx+Cx?+Dx*4 &. =0 
or 
x(B+Cx+Dx? + &c.) =0 
But x being indefinite cannot be equal to 0; .*. 
B+Cx+Dx?4+&.=0 
Hence, as before, it may be shown that B = 0, and therefore 
x(C+ Dx + &.)+0 
Hence C = 0, and so on throughout, 
