ALGEBRA. 
Then, in the triangle SQF, s : p :: z : 
pz 
P z 
: OF, hence 
GM—GF-\-FO-\-QM—d-\- -p v; and, in the triangle 
pz 
MGP, i : s :: j-w : sd-\-pz-{-sv~PM=.y. 
Alfo, in the triangle MKQ, m \ a :: v : —-KO, and SK 
m 
—SO — KQzzz —~; again, in the triangle SKE, i:m :: 
qv 
WL/ 
: mz — qv—SE—DP‘, hence f -— mz — qv—CP—x', 
and, if thefe values of xand^be fubllituted in the equa¬ 
tion which reprelents the relation of CP to PM, an equation 
is obtained which reprelents the relation of SQ to OM. 
Cor. i. Since the values of xand_y are reprefented in 
fimple terms of z and v, the equation to the curve will 
rife to the Tame number of dimenfions, whatever abfcilTa 
and ordinate are taken. 
Cor. 2. From the principles of trigonometry itappears, 
that in, n, and s, may be found in terms of p and q ; there¬ 
fore in the values of xand.y, before obtained, there are 
only four independent invariable quantities, d,f p, and q. 
Cor. 3. If the curve be.a conic feftion whofe centre is 
Cand axis CP, then a 2 y 2 =zb 2 X« 2 —* 2 ; and, fubffituting 
for x and y their values, we have a 2 X sd-\-pz-\-sv l 2 — 
l 2 y^aP—f—mz — <l v ' 2 i or, arranging the terms according 
to the dimenfions of v, 
s 2 a 2 \o 2 -|-3 spa 1 \zv-\-2S 2 da 2 \v 
•\-q 2 b 2 ) —2 mqb 2 J -\-zfqb 2 J i 
-m 2 b 2 '\z 2 -\-2Spda*}z-\-6 2 d 2 a 2 j °* 
+p 2 a 2 J zfmlf) J 
Cor. 4. The equation obtained in the lad article may 
be made to coincide with any equation of two dimenlions, 
Av 2 -\-Bzv -y Cv Dz 2 -\-Ez-\-F— o, by equating the coeffi¬ 
cients of the correfponding terms; becaufe we ffiall have 
lix equations to determine the fix independent quantities, 
a 2 , b 2 , d,f, p, q. Hence it follows, that every equation of 
two dimenfions belongs to fome conic feftion. 
Having given the equation which expreffies the relation 
between the abfcilTa and ordinate, the curve may be de- 
feribed. For, any abfcilTa being affirmed, the correfpond¬ 
ing values of the ordinate are known from the equation; 
and thus, by affirming different values of the abfcilTa, the 
curve may be traced out. 
If ay—bx-\-cd be the propofed equation, it belongs to a 
right line. Let the abfcilTa 
be meafured from the point 
A along the line AB, then, 
^g-when x—o, we have_y=i—; 
the curve has two infinite arcs lying the fame way from A; 
but, when x is negative,becomes impollible; therefore no 
part of the curve lies the other way. 
Let xy—ab ; then when 
and when x is politive 
and very great, y is poli¬ 
tive and very fmall; 
therefore the curve will 
have two infinite arcs be¬ 
tween the lines AE and 
AB\ alfo, when x isne- ■ 
gative, y is negative, and 1 
when infinite, y is infi¬ 
nitely fmall; when infi¬ 
nitely fmall ,y is infinitely 
great; therefore the 
curve will have two infi¬ 
nite arcs between Ab and 
AF. Thefe lines EF, Bb, which continually approach 
nearer to the curve, and whofe diltances from it become, 
at length, lefs than any that can be affigned, but which 
produced ever fo far do not meet it, are called AJymptotes. 
x is very fmall 
E 
1 
P 
y is very great, 
Nii 
__ 
A 15 
M\ 
] 
F 
Let x* — cdx‘ 1 -\-My'— o; then y—dz-^J a 1 —x 2 ; and when 
a* is nothings is no¬ 
thing, or the curve 
palTes through A, the 
point from which x 1 
is meafured. When 
x—±a, then y— o ; 
therefore the curve 
paffes through B, and b, fuppofing AB—Ab—±La ; but,-if 
x be greater than_«, y becomes impoffible ; therefore no 
part of the curve lies beyond B or b. 
To find the conic fection to which any propofed quadra¬ 
tic equation belongs. 
l.<zt ay*b +cx.y-\-d-\-cx-\-fx' 1 —o, a general equation 
of two dimenlions, be the propofed quadratic; then, 
b-\-cx d-\-ex-\-fx 2 , . , 
y 2 -j- - - Xy = ---—-; and, completing the 
from A therefore draw AC at 
right angles to AB, and equal 
and the line which belongs to the propofed equa- 
a 
tion mull pafs tlrrough C. Alfo, if y— o, then x— 
■——; take, therefore, upon the line ADP, AD— —, and 
the line to which the equation belongs mud pafs through 
D ; therefore DCM is that line. 
Cor. If AP be taken to reprefent any value of x, and 
the ordinate PM be drawn parallel to AC, PM will repre¬ 
fent the correfponding value of y. 
Let ax—y 2 ; then, when x—o, we have y— o, or the 
curve palTes through A ; when x is politive, y=dz ax , 
and w hen x is infinite thefe values are polfible; therefore 
r „ b-X-cx b-\-cx\ b-\-cx 
fquare, y 2 -f Xy -[- ■■ r - j =2- 
a 2a | 2a 
b 2 -f- 2 bex -p c 2 x 2 —4 ad — \aex —4 ajx 2 
4a 2 
d-\-rx-\-fx 2 
x ‘ + 2ic ~ 4 gjX andj extrafting the 
4 n 2 
iquare root, y-\ -— 
^0—4af. x 2 +' 2 be — 4ae. x-\~b 2 — 4ad . 
— b cxdz v p — xaf.x 2 -\-2bc — 4 ae .x-\-b 2 — 4ad 
Hence, 1. If c 2 — 40/ be pofitive, when ±x is infinite, 
y has four polfible values ; therefore the curve has lour in¬ 
finite arcs, or it is the hyperbola. 2. If c 2 — 4 a f— o, the 
curve has only two infinite arcs, becaufe when 2 be — 4 ae.'x 
becomes negative and greater than b 2 —4 ad, the values of 
y are 
