797+] : Mathemat cal Phat. 21K 
AEB! ee to ult rites E8, nas x EB! refpetiively ; but thefe re€tangles are given, there= 
? 
‘fins ’ fin. @ 
fore L x ne and 
Ln. Ooty Sno! 

ee x EB’=R’. From ‘thefe equations we have fe x (EB+ 
cera fin. p 
ae é 
EB)=R +R’, and fsx EBX EB/=RR’; but EB+-EB’=2P, and EBxEB/=Q, therefore 




R--R’) fin. RR’ fin. o° 
a= RRs and i $B Lie Whetice eS Gara (== = g > .and: the 
“fin. @ 2 D 
i (RR) fin. © RR’ fin. 9? (R--R’)? fin. 9? 
Fequired equation 27. ————-— cnn =O) allo Pp? = 
q q 7 +: BE eet a 
RR’ fin. Oa rn fin. o? Gr i fin. 9? (Bim —Q) es (R—R’) fin. me 
jis be « 4? 2fe ' 
—R!) fir Gn 
and 2 == SS AU £2. o4 (RoR) fits 0... Roel & Rin. 9, S ot 2 2 Ps that is 1S) seme penuh. P and 
Rie 2p 2p 1G L 
EB! art : . 
Z : \ 
Cor. x. If the relation between ED, DB be fee let ED=y; DE then will fin. ¢—= 
y (R aa fin. o RR? fin: © a HR ono ige 
=? 4? ty?==%", and 27— 

A ee ty Sane : 
» (x2y-Ly3) ee « y?==9, a curve of the fourth order. 
fp 
R R fin. D 

<0: that is, (a2y2)? ae 
RL 
p 

Cor, 2. Let Q==o0, and R’ will be ==0 ;. then z=2P = -==EB, and EB/—=o, that is, 
Eis a point in the curve required. In this cat? the curve is a circle, whofe diameter == —. 
This is the property of the circle which originally fuggefted the problem, and from which oe 
following local theorem is derived, as enunciated by Simfon (Opera Pofthuma, pag. 327): «* Si 
a centro circult O ad rectam AC ducts perpendicularis OG circumferentiz occurrens in H’ K, 
cf per punétum H utcunque ducatur 1ecta A’H’, que occurrat recte AC in A’ et circumferentiz 
rurfus in H’, re@tangulum AH’, H’/H! quale a rectangulo GH’, H/K.” 
Cor. 3. The moft dieukabie property of the curve BB’K, and which follows diedlly from the 
Hy pothefis, is, that the fegments EB, EB’ are in a conftant, given ratio. 
Problem If. Fig. 2. 
To find the curve ABB’, fuch, that if, froma given point E, any right line ABB’ be eae 
meeting the curve in two points "BBS » and if BB! he bifeéted in D, and the perpendicular DC be 
drawn to meet a right line, given by pofition, and pafhng Sucve E, in C, the right line BC 
joining the points B, C, may be given. 
Let the angle CEB=0, and let P, Q be two functions 
of this angle, which are to he cogent from the nature 
of the curve; then, if EB be taken==z, the relation be-~ 
tween x and. may be expreffed by the equation x?— 
2P2-+-Q=-. Now, by the nature of equations, EB-- 
EB! is==2P, or ED==P; and EB x EB’==Q, whence 
(EB--E, B)2—aEB “EBS 4 @Q), or EB’—ER = 
BB’=2BD==24/ (P?—Q), and BD==P?--Q. But DC 
==ED x tan. g==P tan. p, and BC, (by hypothefis) = a 
conftant quantity == R, therefore BD? Toe P2—=Q)- 
P2 tan. 9?==BC*=R?, Q =P? (1-+tan. 97) — R? = P2 
fec. ¢?==R?, and z7—2 Pz (P? fec, » 2am?) —=0, which 
an{wers to an infinite number of curves pofleffing the 
property required, 

A 
Gor. x. Let P= Gee » Where A =a conftant seal then will the equation of the curve 
become z?—2 Ag cof. gp A2—R*=0, which exprefles a circle whofe radius — R, and center oH 
and the diftance EC—A., 
rf 
Cor. 2. If the abfolute term of the general equation be puts, we will have P=-+- a = 
fec. o 
HER cof, g, and 2*-L2R-z cof. p==0. Whence s==0, ort aR cof. ¢, which fhews the curve; 
im this cafe, ta be a circle whofe radius == R. \ 
Montuty Mag, No. XV, , Ee Care. 

