the Infinitejimal Calculus. 34 1 



I know very well that I am committing an error, and I put 

 the equations, mentally, fo to fpcak, into this form, 



and p 1 being fuch quantities as the former equations want 



to render them exact. In like manner, in the equation 



TP y 



Yjp — " — -"-1 refulting from the above two imperfect equa- 



TP 



tions, I undcrjland the quantity <p" } being fuch that (- ... 



' — ] ■+■ <p" — o, may be an exadl equation. But I 



know well enough, that this laft quantity <p" is equal to zero ; 

 or, at leaft, that it is only an infinitely fmall quantity, fince 

 no inrinitefimal enters into the firft term. Now this cannot 

 happen, unfefs each of the terms, taken feparately, be 

 equal to nothing; whence I conclude, that I have exactly 



TP y ... 



•=-== = — — ; fo that the quantities <p, q>* and <p" have not 



been fuppreiTed as nullities, but only tmdcrjlood, in order to 

 Amplify the calculation. 



Again : if A^, for example, be an arbitrary quantity, which 

 may be rendered as fmall as we p]eafe, and if there wer-e 

 given an equation of this form, 



A + BX + CX* + &c. = o, 

 A, B, C, &.c. being independent on A", this equation cannot 

 exift, unlefs it be A = o, B = q, C = o, &c. ; that is, 

 unlefs each term, taken feparately, whatever be their num- 

 ber, be equal to zero. And, for the fame reafon, if we have 

 an equation of this general form, P+ «!fj,= o; fo that P 

 may be a function of the quantities given or determined by 

 the conditions of the problem ; and, on the other hand, Q, 

 ji quantity which we may fuppofe as fmall as we pleafe, we 

 fliall neccflarily have P = o, and Q = o. But fuch is pre- 

 cifely the nature of the equation in the laft article, namely, 

 ( TP _ y \ ( TT _ zy* • (RZ ■. MZ) + aRz-x iiZ\ _ 



\ y a - x) \ y (a — a) . (2a - zx — MZ))~°' 



Therefore each of the terms of this equation, taken fepa- 

 rately, is equal to zero; and confequently, the quantities 

 T'T, MZ and RZ } which enter not into the firft term, may 



Vol, VIII. Yy be 



