ISOPERLMLTRY. 



{(+<ty) ^'.v. :■ 



and tlicrcfoiv 



(• + P'-) 



■*liic!i exprefiioii, when c — o, becomes 



^'' ('-"'->') which, by redudion, becom 



■whence the rekilive values of a- and y arc determined in 

 both tai;.s. p, or ^ 



Pk03. VI. 

 Required the folid of leail refiilance, amonjft all tlii ^ " 



which, if ;: = o, reduces to 



^. _ - y' 



--^'-—i 



('+/>=)* 



fuhds oi equal capacity 



. Here, tlie fame as in Prob. IV., we have V = — ^ 



rry---c 

 ,/ [a' - {-y' - c)'j ' ' 



we iiavc 



and fti X = f-v'^x, (- beinsf equal j.ijkq, &c.) (fee -m ■ i /i r i • . «• 



Woodhoufe's-' Iloperimetry, p" .a^^ ; therefore V - u. . ""^ ^^^ ''^^^' %''' ^"1' S.mpfon's, but it i, reftricled, 

 *^ ■' ^ -' ' becauie c is not ncceirarily equal to o. 



Phob. VIII. 



The length of a curve being given, it is required to deter- 

 mine its nature, when the area is a maximum. 



Here, by proceeding as in the foregoing examples, we 

 fliall have 



V' = 



yp 



-~ — -_ — ay" (including t in the quantity :i). 



H.nce P = -3.V/'__iL-'';L' and by formula [«] 



I -r/.- 

 vhence, bv reduclion, we obt 



5yp'+yp' 



and therefore. 



S'c-ay"-) (I -^f-y = 2yp\orl 

 \{c — iiy'.tlz' — 2yj'x j 



<ftiich expreffions will furnifo the relative values of x and ji. 

 If, in'tead of the condition of equal capacity, that of equal 



fuperficies be lubilituted, we have 



V *^ 

 V — <ja = V = -^i ay ,/ (i + ft"), and 



V = ^/(i +p-) 



and hence, by means of the formula [a], 

 Tliereforc, 



y (i +/■')- ay 



(I +/ )' 



"yp 



vMi + P) 



V' -r P) 



•whence we deduce from formula [^z] 



/:* ,. - "y N^ (i +/■') = 



I - p 



sy p^ + y p' _ "y p' 



U+P'Y .ni-rp) ' 



'which, by reduction, becomes 



• c [I +py- = 2yf>^ + ay {l + f)) 



?r, ci* = 2y_i' .i + ay iKi. ■ 



Prob. VII. 

 Given the length of a curve, to determine its nat 

 when the folid generated by its rotation is a maximum. 

 Making, as before, rr r=. 3 I4t59, &c. we fiiall liave 



/Vi=/^_v^v,and/«.^ =// (. +p^-)i. 

 Therefore W orV — au — -zy — a v' ( i + /') ; 

 and talking tSie fluxion on both fides, 



v'-2-.. ±t>L-^ ■ 



•comparing this exprefTion with the general one [A], 



v.'c have P = — , ' , ; an 



formula Va] 

 Vol. XIX- 



vhich, being farther reduced, becomes 



/,=!=, '' Ti - {c + ay)-} 

 ^ A- {c+ay) 



,^ \_l -[c ^ ay)'-'^' 

 the fluents of whi^h being taken, we have 



x^c - ,^ll~ {c + ayy--^, 

 which is an equation to a circle. 



P.fOB. IX. 



Required the curve that generates tlie folid of the leaii 

 furiace, the area being given. 



Here \ = 2-z y ^' (\ ■\- /.-), and u = y ; 

 therefore V, or X — au =^ z -^ y ^' [i ^ p'^ — ny, 



and V = [I ^ ^/ ( I + /-) - «] v + -^:^^'-, p ! 



\' il -V P) 

 whence, by comj^arifon with formula [A], 



v'(l +/-)' 



then, again, by formula [j], we have 



2.y..^^+p')-ay = -^£^+., 



V (I +/) 



, , , , which, by reduftion, becomes 

 d, theretore, by . 



P = 



c +■ ny 



