4? Gzr^oif on (h Tbupyy of 



I. Since tiie difTcrential of ,v ij; dx^ the integral Bf Aic will 

 t)c .v; that is, we fliall have/^Av = .r. But, as the differen- 

 |i|gkl of <j + jf is ^Ifo <^r (by article 5.^) it follows that the in- 

 tegral rtf r/r is at> juflly cxpreiVcd by ^ + a-, as by >i alone; 

 and that, in general, every dift'erential hath as many various 

 integrals as we may choofe to afli^n to it ; but that aU thefe 

 integrals only diffei* by .i eohftant quaiitity. It fuffices, there- 

 fore, to determine one of them, and to add fome conftant 

 (|uantity to reprci'ent all the reft \ that iS, all the poflible in- 

 tegrals of dx will be rcprefcnted by .v -f- A^ the quantity A 

 being a conftaiU quantity, takeii at pleafufe; 



t. Becaufe the differential of a; +^ + 2; &c. is dx + dy -^ 

 dz Sec. the integral of this differential will be ^ + y + x; &c. 

 -hA. 



3. The differential of xy being xdj^ +j;dx (by article ^6) 

 as well as that of xy -)- A* the integral of xdy ^jdx. will be 



i\\ . : *"f :~ V' • 'o •..'■•'■■^' J '■ ■ i -■■'■i '^ '■■■■ ■ *' 



■teciprocaUy x}' + ^J^ 



VtlV ~~ Xu\' 



4. In like manner^- we fliall find the integral of — — ~^ 



to be — h A\ 



- y . 



ry. So likewile we fliall find that the integral of w.r "* dx^ 



15 X + A, &C; 



Such are the principal rules of the Integral Calculus. We 

 proceed to fltow, by fonie particular examples, the applica- 

 tion of tiiefe rules, and of thofe of the Differential Calculus ; 

 both which we, fliall do as fuccin6lly as poffible. 



Application of thcfe general Vrinctples to fome 'Bx ample s^ 

 59. Ah elliptic curve AlMB (fig. 3.) being given to find 



TDtift have recoude to more extenfivc works, fuch as the Fluxion^ 

 rf SimpfoTi, Emerfon or Maclaurin, jlnd above all, thofe of Ditton or 

 l-'HofpltTi], as tr^nilatCH^ ahtl augirtciitccl by Stone • which two lart, cfpe- 

 tlally Ditton's, are generally confidcred as thfc plaineft works on the fub- 

 it^. The perfpiculty of Sihipron^s excellent tra^, in his Scle£t Mathe- 

 matical ExercifcsV has been already 'mentioned. The fame excellent 

 ■qU;ility pervades the fix DilTcrtatibns on tKe progrcfs of Geometry, in- 

 frrtcd in ** The Mathematician/' printed iii 1751^ of which tke three 

 liH arc' <!c)n fined to Fluxiuns.— W. D. 



4 ihe 



