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 algoridim, by 



dy 



"In the same manner 



d(dy) 



or 



the Indefinite Difference of the Indefinite Difference of y, or the 

 second indefinite difference of y. 

 "Proceeding thus we arrive at 



d"y 



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 PKINCIPLE. 



"It is evidently a truth perfectly axiomatic, that no aggregate of 

 INDEFINITE quantities can he a definite quantity, or aggregate of 

 definite quantities, unless these aggregates are equal to zero. 



"It may be said that (a — x) + (a -f x) = 2 a, 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^+Dx^-f&tc. = 

 A, B, C, &:c. be definite quantities and x an indefinite quantity; 

 then we have 



A= 0, B=0, C^O, &£c. 



" For Bx + Cx^ 4-Dx^ + &ic. cannot equal — A unless A = 0. 

 But by transposing A to the other side of the equation, it does = — A. 

 Therefore A = and consequently 



Bx + Cx^+Dx^+fcc. = 



or 



x(B + Cx + Dx-^4- &c.) = 

 But X being indefinite cannot be equal to ; . ' . 



B 4- C X + D x -^ + &£c. = 

 Hence, as before, it may be shown that B = 0, and therefore 



X (C + D X 4- &c.) + 

 Hence C = 0, and so on throughout. 



