tloeJnJlniteJimal Calculus, 53 



.For example, let the equation of the propofed curve be 



hy = ax^ — ;r% th^ diflferenlial of which Avill be the 



imperfedl equation, h'^dy = laxdx — '3,x^dxy 



f dy . 2ax-^3x^- 

 or the accurate one, u?n. [~,;j = ji 5 



theretore j^ is a mimmuytiy 



or Im. \— ■ — —^\ = o, 



^ ax / 



and hence we have, 2a — 6x — c, or x = ^a*. 



66. Frohiim IV. To find the area of a parabolic feg^ 

 iiient. 



Let AMP (lig. 6.) be that fegment : if we fuppofe the 

 abfcifle AP to be increafed by the infinitely fmall quan- 

 tity PQy the fegment will increafe, in the fame time, by the 

 quantity MiYPQ; that is, PQ being fuppofed the differen-. 

 tial of x^ MNPQ will be the differential of the fegment whofe 

 iurface is required. Converfely, therefore, that fegment is 

 the integral of MNPQ ; that is, AMP = /{MNPQ) . But, 

 letting fall MO perpendicular to NQ, it is evident that the 

 ultimate ratio of the fpace MNO to the fpace MOPQ is o. 

 The former fpace, then, is infinitely fmall, compared with 

 the latter; ai^d hence we have the imperfed equation MNPQ 

 = MOPQ. Subftituting, therefore, the fecond of thefe 

 quantities for the firfl;, in the accur^ite equation, AMP = 

 J (MNPQ), we (hall have the imperfecl: equation, 



AMP :=r /{MOPQ), or AMP ■^- fydx. 

 But, calling thp parameter of the parabola P, the equation of 

 that ciirve is 



yy c= Px\ whence dx — —77- , 



an impcrfefl equation. Subftituting, then, for dx^ m the 

 iirfi: imperfeft equation (AMP = fydx) its value in the 

 fecond, we Ihall have this ijew imperfe(^ equation 



AMP = /^. But /"-^^ = ^ (by article 58); 



'* In this {blution, the author, or rather perhaps the printer, had by 

 miftakc put minimum for maximum^ anJ maxi?num for minimum ; but 1 

 Jiave made tlie ncft ff^ry alterations. — W. D. 



and 



