the hijlnitcfimal Calcuhis, ^*f 



winch, like them, lead, lieverthclefs, to rcfults rigoroudy 

 €xa6i * . 



57. By proceeding' as in the kftaniele, we (hall find the 

 imperfed: equation 



d ,xyz = xydz + xzldy + yzdx» 



In Hke manner we (hall find the imperfe«Sl equation d,x 



= Tttx dx bcc. 

 And, by the fame kind of procedure, wc difcover the im- 



perfect equation d, — = .- -. 



/ . ^^ 



58. Such are the principal rules of the Differential Calculus. 



Let us now proceed to thofe of the Integral Calculus, which 

 is the Invcrfe method f. 



I. Since 



* If from the imperfett equation d ,xy ==■ xdy\- ydx, 1 wifhed to 

 derive a rigorouUy accurate equation, I might do it at once, by re- 

 ftoring the terni dxdy, which it wants. But I might alio effe6l it 

 in the following manner. Dividing the preceding equation by dy, 



for example, I obtain the ij^w iraperfe<ft equation '' - = y -— 



dy dy 



4- X ', and as (by article 19) an auxiliary quantity differs infinitely 

 little from its limit, I may fubftitute in this lalt equation Htn. 



(— ~) inftead of ——- itfelf, and Jim. ( — 1 inftfead of ~ it* 

 *fy J dy ' \dy J dy 



ilelf. Now, it thence becomes lim. l-^~\~y x lim.l-~-\ -f x. 



But every limit is (by article 17) an alfigned quantity j therefore, 



though dx and. dy are, themfelves, auxiliary quantities, lim, f • ' ' j 



and lim. {-~- ) are affigned quantities.. All tbe terms, therefore, of 



the preceding equation, /;/«. (— p-) ~ ^ X lim. (-7-) + x, arc 

 \ ay * ^ tly I 



afligned quantities ; and confequently (by article 34) this equation 



is neceffarily and rigoroudy exa«f^. 



t The author touches the practice of the Intei'/al Calculus as (lightly 



AS he does that of the Diiferential » but t^e furmcr does not fo dire<5try 



bear on his fubje^l as the latter; and, if it did, I could not expedl to bd 



indulged with fo^long a note as I have given on the other; and which, 



aster all, is but a flight and impcrfeft general outline. For particular in- 



fow(ftions, the reader, who is ac<juainted with the nectlTiiry pneco^f^iiaf 



mufl 



