ISOPERIiMETRY. 



muttlply this by/, and fiiice -4- = ?, we thus obtain ^hcn will V = c p + R r - j 1 + c [/] 



T) ; ,\ . ; By nierins of which formulx the following problems are 



Vp = gQ + cj>;_ readi'ly folvcd. °^ 

 and now fubftitutiiig tins value of P p, in the general ex- 



preffion for V, it becomes Phoblem I. 



Y _ Q .|_ Q ■ ^ ^ / Required the relation of x and y, fiich that the fluent of 



,, , I- .in , u ?! n (" "^ — J' ) J' -'^» or f(a.v —y')y X, fliall be a maximum, or 



and by takmg the fluents on both iides a mininium •'''•'' ' 



V=Q? + <^/> + <:'; Comparing this cxpreflion with /V a-, (the analytical ex. 



f and <■' being the cojredions. prefiion for tlic maximum or minimum property,) we fee that 



If M be not equal to o, we mud add the fluent of M v ; V = a.ry — y', confequcntly V = ayx+ (.i .v — 3 j ) i\; 



in which cafe which compared with the general formula, 



V = fU x + Q q + c p + c' V = M -v + N .■ + &c. 



Cafe 3.-If M = o, but N is not = o ; that is if the ;.,.^.^ ^^ ^ ^^^ N = «.v - 3 f, P = o. Q = o, &c. 



form be ' p .' ' ' x, ' 



V = N V + Vp + Q q; ^"' '"'"ce N r + &c. = o, this becomes 



r o ' '- ' — ' 



then fince N ^ -i- -?r = o> a *■ — 3 J" — o, or _>•=./__; ; , _ 



.XX- ■ ^ V ^. 



, , V 1 • u ■ * • /"i^^, ,r A ' ^ which is the relation of A- and )i required, . ' 



we have by multipiving by _v — p x, ( becaule p = -- ] j ^ . 



PaoB. II. 



"N J — /> P + p -.- ■»' = o- Required the (horteft; curve that can be drawn between 



■* . . two points, or between two curves.. 



Q - / Q \' ■ ^^ _ /' A ^ \' _ ^^'■'■''' '^ "'^ '^'^''" '^ '" reprefent the curve, k the abfcifs, 



^ Ejt/> -r -^ = \P .;. ) ~ P '^'^ yP V/ and _>| the ordinate, it will be neceffary to fulfil the following 



condition, namely, that f x, be a minimum, 



Q ,,,„ ^ ,■ rin-.jr.y. ^"^ W the doarine of fluxion?, z = »/ ;.' ■+- i'- — 



q 7-.;, whichla.l expreflionbemg fubdituted for/. -^ ,y, •' ' ^ ^ "t" 7 



Q\ 



gives ^y = p^- [f^:) 



^ \ / ia.2L = ^/v/i4-A'; which being compared 

 with / V X, gives V = ^' i + /> , whence by taking the 



V=/,P+P^— (/I -^^ + J Q + Q ? > fluxions on both fides, it becomes V = -^-^^ ; and 



and taking tlie fluents M, N, &c. all equal to o. Therefore comparing this with 



V= 'Pp+Qq+p^ + c, *'^^" gmexaX value of V, we have P = -_--f?--- and con- 



f being the correftion, as before. ^ fl-quently from the formula [B] we have, _ flux. 



We might continue thus to deduce from the general for- / /> \ . ^ , 



mula other particular ones, but the foregoing are fufficient \':;;r(i'::;rj') ) = °5 tnerefore / = a ,/ (l + /. ; ; and 



for the folution of fuch problems as the limit of our article , C„,,^,.\,Z. a-- _,,-.,„,/ - , w _ - i 



will admit of in tlxis place, which, for the fake of a more ^y iq"a""p = a + a p , or [a - i) p = ~ a , whence 



ready reference, we will again repeat. /. = 4 == —~- ; or >' ^/ (l — aM = a i. 



If thegeneralform V = M * + N v + P/ + Q 7 ; 



taking the fluents v ^' {i — a) = « .v -(- c, which is 



becomes V = N y + P /> equation to a risrht "line, as it ought to be. 



then we have V = P /- + r [a^ 



or V =fUi+V p + c lb-] PuoB. III. . 



when M is not equal to o. -Requiredthe curve of quickeft defceot between two giveo 



2. If the formula be V = P /. + Q 7 points. Fig. 16. Plate Ijopcrlmetry. 



then will V = Q 9 -f r P + <:' [.] Let A and B be the two given points, and A M B the 

 or V —J'M x + Qq + cV+c' [r/j required curve ; drav/ PM perpendicular to A C ; and put 



when M is not equal to o. " ^^ ^ T •^' ^ ^' "" r''' f^^^^ ^'- *'''"' ''"" .'"^ '""«^ "•''! 



. . . be rcciprccally as t!ie Iquare root of the height, it will bo 



3. If the formula be V = N j- + P/ + Qy ,1^ 



• required to find / -^^, a niinimum, but x = ^/ (i'- .). 



then will V ^Tp i- Qq -p y- ■{■ c [f] -' ^'-^ 



To which we may alfo add the following, j)'), whence / ~-i— - ^-' ' =: C^'^^'^'J'I v = 



4. IftheformulabeV = P^'f Rri *■'' V\j' 



