A L G E 
Let QONM be the curve, AP the abfciffa, PQ an ordi¬ 
nate, meeting the curve in the points M, N, 0 , Q, and the 
Let ABP be an abfciffa, and from Mand D draw MP 
and DB at right angles to ABP, and let y n — ax\by n 1 
. -\-px n '-\-qx n ’+ &c. y-\-Px n -\-Ox n ‘-p&c.reo 
be the equation which represents the relation of AP to 
BRA. 323 
PM. Alfo, let AC—z ,CM—v, CBzzzo,BD-u ; S./^MCP 
—s, S./_CMP-^zc, to radius 1. Then 1 : s :: v : sv==y;. 
,ind j : c :: v : CP, hence xzzAPzzez-\-cv. If thefe 
values of x and 7 be fubftituted in the equation to the 
curve, the relation of AC to CM will be known; and the 
coefficient of the laft term but one of the transformed 
equation, divided by the laft term, will be the fum of the 
reciprocals of its roots, or ----&c. 
Now, fince x~z-\-cv , 
x n —z n -\-nz n 'ev 
x n —'—z 7l —'-\-^.z n — 
CM CM 
afymptotes in a, b, c, d, then PM-\-PN-\-PO-\-PQ—Pa-\- 
Pb-\-Pc-' r Pd, and by tranfpofition PM — Pa-\-PO — Pc— 
Pb — PN-\-Pd — PQ, or aM+cO=zNb+Qd. 
Cor. In the common hyperbola MCN whofe centre is 
0 , and afymptotes Oa, Ob, if any line aMNb be drawn 
3 cv-\-n- 
It • 1 
■ i 
1 
—U — 
I 
- n- 
V o 2 -j- Sec. 
Sec , 
— z n Va 3 -f- See. 
See. See. 
and, fubftituting thefe values for x and its powers, 
and sv for y, in the terms of the original equation, we 
have the two laft terms of the transformed equation, 
pz n ‘+<72" °-p&c.x sv-\-nPz rl — i-Qz’ 1 2 -j- &c. 
X cv, and Pz n -\-Qz n See. all the other terms involving 
the Square or fome higher power of v ; hence 
s X pz n ‘-f qz n See .-f cx nPz 11 ‘-f n — i.Qz n 2 +&c. 
~CM + ‘ 
Pz n +Qz n *~'+ See. 
„ 1 1 
-&c.: 
CD 
and 
CM CM 
yj 
sxpz n '-{-qz n t -\-Scc.-{-cxnPz n *-J-»—1 .Qjz n s -j-& L '' 
cutting the curve in M, N, and the afymptotes in a, b , then 
aM is equal to Nb. 
If a ftraight line DCM be made to revolve about C, and 
/ // 
cut the curve MMM in as many points as it has dimen- 
lions; and, if be always taken equal to -|—~ —. 
CM 
— -&c. the locus of the point D will be a ftraight 
It 
CM 
line EDF. 
-.y.Pz n 4 Qz n —' ‘j-jpqtk 
y/ uP -J-iP 
Alfo, from the fimilar triangles MCP, BCD, 
--- u 
yj 
yj ad-J-a 2 : zvi: c— 
V u?-\-uA 
w 
yj zv 2 +a* 
Therefore 
yj lw*-f U 2 
XnPz n * 4-72 
:Xi t>z n l +qz n 1 +&c.-f 
Qz n ~~ 2 + See. — 
yj w‘-\-U 2 
yj UpP^vA 
- x 
Pz n -\-Qz n ‘‘-J-&C. or uXpz 11 l -{-qz n *-\-bec. -j- w X 
nPz' 1 — i.Oz 11 2 -|-&c. — Pz ,l -\-Qz n Sec. and, 
fince the point C is fixed, AC or z is invariable ; there¬ 
fore the relation between w and u, or CB and BD, is ex- 
preffed by a fimple equation, and the locus of the point D 
is a ftraight line. 
In the general equation jy 71 — a^ r bx.y n ‘4- See. —o, if as 
be fo allumed that two roots are impoffible, two values of 
the ordinate belonging to this abfciffa are impoffible, that 
is, there are no lines which reprefent them. Hence it is 
evident, that in deducing the properties of the ordinates 
from the equation to the curve, we nnlft fuppofe all the 
roots of this equation poffible; becaufe, though the Aims, 
powers, products, Sec. of fuch impoffible quantities may 
become poffible, and their relations, difeovered by an al- 
•gebraical procefs, may be expreffed by poffible quantities, 
yet the reafoning does not extend to curves, in which the 
original quantities cannot be reprefented. 
On the fubjeff of Algebraical Curves the reader 
may confult Dr. Waring’s Proprietates Algebraicarum 
Curvarum. 
ALGEBRA'IC, 
