582 
A SIXTH MEMOIR UPON QU AN TICS. 
[158 
206. The line-coordinates (£', y, £') of the axis of inscription are 
ax' + hy' + gz, hx + by' + fz, gx + fy' + cz\ 
and we thence deduce the relation 
V. ty = K(a,...W, y’, ¿f. 
In order that the form 
(a, y, zf (a, y\ z'fcos 2 0 - {(a, y, z\x, y, *)} 2 = 0 
may agree with the originally assumed form 
(a, y, zf + X(%x + y'y + %'zf, 
or what is the same thing, 
(a, y, zf + X {(a, ...$#, y, z\x\ y', z)) 2 = 0, 
we must have 
x = ZL 1 
(a, ... ][x', y', z'f cos 2 6 ’ 
which may also be written 
, -K 
v, rO’cos^' 
or what is the same thing, 
K + V, n 2 -^(a...5r. V, H 2 sin 2 0 = 0; 
and we thence, by a preceding formula, obtain the line-equation of the inscribed 
conic, viz. 
207. The point-equation being 
y, zf (a, ...$V, y\ /) 2 cos 2 6 - {(a, ...fa, y, z\x\ y', /)} 2 = 0, 
or 
(a, ...][x, y, z'f (a,... $V, y, zf sin 2 6 — (j?t,. ..\yz' — y'z, zx' — z'x, xy' — x'yf = 0, 
equivalent in virtue of 
(»,•••$>, V, zf {a, ...fix', y', z'f- {(a, y, z\x\ y', z'ff 
= (8, • ’-\yz - y'z, zx' - zfx, xy' - x'yf; 
then the corresponding forms of the line-equation are 
№,-№ v , sm-iF. v', cyan's-{(&,...■№, y, f$r. F, r)) s =o. 
V, V g'-zt iy-i'yf= 0, 
equivalent to each other in virtue of the before mentioned identity 
(».-$?, % rm v. o ! -r$r.r» 2 
= (a, fV-F’/)“-