ISOPERIMETRY. 



/ ' *, a minimum, which compared with the whence, x = „ — + + — hyp. loir, h + c' ; 



J Vy Hp"* 2f 2 '^ ^ ^ ' 



minimum expreflion/V x, gives V = ^^ ^' "^ ^'^ ; whence ''>' """''"' °[ "I.ich equation, and that above, -vh. cU+ /.'>' 



^/ ^ ^ 2yp , the relative values of x and _y are determined. 



V = - ^ilLtn y + ^ , ,,hieh com. Pho«. V. 



pared vrith formula [A], gives M = o; N = — Required the bracliyftochrone, or curve of quickeft Ac- 



+ /'). p / 



fcent, when the length of the curve is gi' 



, P = —— -L _; Q^ -— Q^ g-j.^ This is an ifoperimetrical problem of the fecond order: 



^ J'-i v y • \/ (' +^ ) thofe which we have been confidering have only one con- 

 Therefore, fince by formula £a] V = P^ + f ; there- dition enters, namely the minim.um ; but in the prefeat 

 . ^/ (i +/>') />• problem, befide the minimum property, the ifoperimetrical 

 tore m this cafe — = — — - — -)- c, alio enters ; and we have, therefore, to find the variation of 



V y -.' y-v (I + /■) 



(V — au), inftead of V. See Woodhoufe's Trad on Ifo- 



whence ;; = cj or multiplying both nu- perimetry, p. 1:2. 



V y • \^ (I + P) Here by means of tlie companion of y (V — au) x, witk 



merator and denominator of this fradlion by x, it be- the minimum condition of the problem, we obtain 



—r — c, wliich is reducible to ^ = V = -111L±^\ and u = ,/ (i + *n • 



r) Vy ^ V -r/-;. 



-J^fzr-^) • ^' '" ^'l"'''"" '° =* '5'cloid ; which, there- therefore V-cu, or V = ^.±t} - a ^ (i + fi\ 



fore, is the curve required. ^ 



PnoB. IV. 



and confequertly. 



V = 



jzr -/ (^ + P) y' + C "TV "~ " ) ^ 



Required the curve which, by its resolution round its • •'' \ \ y -^ 



axis, generates the folid of ieaft refinance. Fig. 17. P_P_ .^ 



Let A BC be the required curve, which by its rotation V (' t- />')' i|^. 



generates the folid, D A C, of Ieaft refjlf ance : draw PM Hence we deduce, by comparing this with the genVd 



perpendicular to AB; put AP ^ J, PM = ^•, and the formula T A], ' f t> 6 



arc AM = a: then we know from the principles of me- 



chanics, that the refinance = / — , which muft 



./ .v^ -'ty' 

 therefore be a minimum. Now this being put under 

 the form. 



./ -v -'ty' 



put under 



Hence again by formula [a] 



1p= (J_ _ ,) X -J 



which being compared with the exprefllon / V x, gives "''* \~y ~ " ) ~ ' ^ U + / ; • 

 Y _ y P . whence again by reduAion, 



P 



the fluxion of which expreflion being taken, gives ■* "" ,/ [i — 2 a ^/ » + (a — - ) v] ' 



y _ p\y iSyp^ "*• y p*)p . where, by taking the fluents, we have the relation between 



~l4-/i- (1+ P"Y " ^ ^"^ y* *^^ abfcifs and ordinate of the curve. 



, , , If, inllead of the length of the curve, the area Iwd been 



whence M = o ; N =^ — ^ • P = •^•^ ■^ "*" ^'P given, then fu x = /y x, and confequently we fliould 



I + /■" (^ + PT ' have, 



Hence by the foregoing form [a], and ufing - c for a W = V — au= ^iilLt£J - av- 



correction, we have ^/ y ^ * 



in which cafe. 



p = 



p 



Vyv{i+fy 



,nd^^±£l_.,= £ - + .: 



whence again, 



which by reduftion becomes, 



