algebra. 
x-\-a 5 
2 — C 2 —b\ 
fides, c 2 — 2<TA;+.v 2 4-7 2 =4a 2 — 4 fl X ^c + *l 2 +^ 2 + ( ' 2 4 * 
2M-j-A- 2 47 2 ; that is, by tranl pofition, 4 « 2 + 4 f * — 
+7 2 , or <z*-frw= aV c-\- 7 l 2 +y 2 ; and again, 
fquaringboth(ides, q*J r 2a i c x-\~i. 2 x~ —a~c 2 J r ia' 2 cx-i r a' ? -x 2 
y~a 2 y 2 , or« 2 j/ 2 =ffl‘ l — o. 2 c 2 —a 2 — f2 X* 2 > let a 
then a 2 y 2 =a 2 b 2 —b°x 2 , and y 2 ——Y . a ~—**• 
Cor. j. If S and //coincide, c=zo; hence d—b, and 
y'—cf — x 2 , the equation to a circle. 
Cor. 2. When x=-{-a, or —a, then;/—o; therefore 
taking CA—CD—a , the curve paffes through A and D. 
Cor. 3. If AP—z, then x—a — z ; therefore,;: 2 — 
1 2_ 1 2 - 
— Xa 2 — a 2 +2az—z 2 =~xzaz — z 2 the equation 
a 2 a 2 
which expreffes the relation between AP and PM. 
Cor. 4. If AS be finite, and SM+MH be increafed 
without limit, the curve at all finite difiances from S be¬ 
comes a parabola. In this cafe .z 2 vanilhes when compared 
b 2 
with 2 az\ therefore the equation becomes v 2 ——X 2 az\ 
a 2 
alfo b % — a' — c l —a-\-c X cl — c, and fince the difference be- 
ween a and c is finite, and a is infinite, c=zza\ hence 
2>a AS 
F—zaxAS, therefore y 2 ~ - X 2az —\ASx z - 
a 2 
To find the.equation to the Hyperbola. Let two indefi¬ 
nite lines SM, 
HM, revolve, in 
a given plane, 
about the points 
S, H, and cut 
each other in M, 
in fuch a man¬ 
ner that HM — 
SM may be a gi¬ 
ven quantity ; 
the locus of the 
point M is an 
hyperbola. Bi- 
fe£t SH in C, and draw MP perpendicular to HS, or HS pro¬ 
duced; let CP—x, PM—y, SC—c, HM — SM—za. Then 
l/ w-j-c~1 2 ^ x — c\‘-\-y 2 —za, and a'yt—f — M .x* 
—Me 2 — a 4 ; in this cafe 2c is greater than 2 a (Euc. 20.1.), 
let therefore F—P — a 1 , then a 2 y l —b 2 x‘ 1 — Mb 2 , or y~— 
b 2 
P9 
Let AB be the dia- 
-Xx —a-. 
b 2 
Cor. 1. The equation to the ellipfe y 2 —~ X a 3 — 
becomes the equation to the hyperbola if Z = be fuppofed to 
be negative. 
_ 
Cor. 2. The equation/— ~xM — may be confi- 
a* 
dered as the equation to any conic fedtion : it is the equa¬ 
tion to an ellipfe when b 2 is pofitive, to a parabola when 
F is infinite, and to an hyperbola when b 2 is negative. 
Cor. 3. If SM — HM—za, a figure fimilar and equal to 
the former will be traced out, which is called the oppofite 
hyperbola. 
Cor. 4. If xz=.±a, thenj = o; therefore taking CA— 
CD—a, the curve paffes through A and D. 
Cor. 5. If AP=lz , then CP or x—z-i-a, and x—a'— 
b 2 —_ 
z 2 -\-zaz\ hence y*z=.— X z * -+■ zaz, the equation which 
expreffes the relation between AP and PM. 
Cor. 6. In the oppofite hyperbola, x— z — a: 
b 2 - 
x°-~~a 2 =.z’ — zaz, and y* = — X. ~ 2 az. 
To find the equation to the Cijfoid, 
meter of a femicircle 
ANB, from the points R 
and P, taken always at 
equal diftances from A 
and /], draw RN, PM, at 
right angles to AB, and 
join AN meeting PM in 
M\ the point M we trace 
out a curve called the Cif- 
foid of Diodes. From 
the nature of the circle 
ARxRB—RN 2 , and by ^ 
the conftruftion AR X 
RB—PBX^P\ alfo from the fimilar triangles APM, 
ARN, AP : PM-.: AR : RN, or AP: PM:; PB : RN — 
PBxPM , PB 2 xPM* „ „ 
and RN —- ——PBxAP ; [ther.: ore, 
AP 1 " 17 AP 
PBxPM*—AP 3 . Let AB—b, AP, 
b — xXy"=x 3 , the equation required. 
To find the equation to the Conchoid. 
given in pofition, ^ 
and about any point 
C, taken without it, ^ 
let the indefinite line 
CM revolve, and cut 
AB in E ; then, if 
EM be taken always A 
of the fame length, 
the point M will 
trace out a curve 
PM—y then 
Let AB be a line 
—— 
/% 1 
J 
£ 
therefore 
which is called the Conchoid of Nicomedes. Draw ADC 
and MP at right angles to AB, and MF parallel to it; lid 
CA—a, AD=zEM—b, AP—x, PM—y. Then from the 
fimilar triangles CFM, MPE, CF (a-\-y) : FM (x) :: MP (y) 
: PE—-^—\ and EM 2 —EP 2 +PM% i. e. b*—~ ** r ■+/; 
a _ a -\~y^ 
therefore, a -f ;-] 2 X P—xy 2 -\ r a-\-yY Xy\ or ~a+yY X P—y 2 
—xy*, the equation to the curve. 
To find the equation to the Logarithmic Curve. If in 
the indefinite line AE, we 
take AB, BC, CD, &c. al¬ 
ways equal to each other, 
and ordinates AF, BG, CH , 
DI, &c. be drawn in right 
angles to AE, and in geome¬ 
trical progrellion, the curve 
FGHI, which paffes through 
their extremities, is called the 
Logarithmic Curve. From J 
the nature of logarithms, any abfeiffa AC is the logarithm 
of the correfponding ordinate CH, in a fyfiern which de¬ 
pends upon the magnitude of AF and BG, fuppofing AB 
given; in the fame fyfiern, let 1 be the logarithm of a, alfo 
let AC—x, CH—y ; then x— log .y, and 1= log. a, or x— 
xx log. <2 '; therefore log. y—xx log. a— log. a x ; hence 
y—a x , the equation to the curve. 
Havinggiven the relation between one abfeiffa CP, and 
ordinate PM, in a curve, 
to find the relation be¬ 
tween the abfeifik SQ, 
which is meafured from 
a given point S in a given 
. diredhion, and the ordi-. 
nate OM, which is in¬ 
clined to PM at a given 
angle. 
Suppofe PM perpendicular to CP ; produce MO and 
DPC till they meet in G, draw SB, SD, SF, refpedtively 
parallel to MG, MP, DC-, and let S-AKSE^p, S-ZKSE, 
or MKO—vi, S. aHMO—lj, S-AMQK—n, S-A^FE, or 
or MGP—s, to rad. 1; SBz=tFG=d } DC—f, CP—x, PM—y , 
SQy=zz, QM-v, 
• Then, 
