MR. A. CAYLEY’S SIXTH MEMOIR UPON QUANTICS. 
81 
or what is the same thing, 
(a, y, zy+X{(a, . y, zjx\ y', z')y=0, 
we must have 
-1 
^ {a,, y', z’)^ cos^ S’ 
which may also be written 
-K 
costs’ 
or what is the same thing, 
-B', 17 ^=0 ; 
and we thence, by a preceding formula, obtain the line-equation of the inscribed conic, 
viz. 
207. The point-equation being 
{a, . .X^, y, z)\a, . .X^, y, 2? cos'* 6— {{a, . .X^, y, zjcc\ y\ 3')}"=0, 
or 
(«, . .X^, y, Zf{a , . .X^, y, zj sin" LJjyz'—y'z, za^—z'x, a^y — a/^)"=0, 
equivalent in virtue of 
(a, .,X^,y,z)"(«, . .X^,y,2T— {(«, ••3(^^'“y^’ zx'—z'x, xy'—ci^y)\ 
The corresponding forms of the line-equation are 
(a . .'ll, ,, l )\%, . .xr, rl, tlf sin" 6 -{{^, . .XI, m, rl, t)Y=^ 
and 
(a, ..XI. ». i)’(a. ..ir. >i, j')’cos*«-k(«, ..x-ii'-o'i, ir-i'i, ix'-i'o)’=-o, 
equivalent to each other in virtue of the before mentioned identity, 
(a, ..XI, ma,-xi'. V, s’f-{(a, ..II, v, nr, r, r)}*=K(<!, ..i^r-rt, ir-«, ir- 
208. Write for shortness 
{a, ..lx, y, z)"=00, 
{a, ..lx, y, zlx', y , 2 ) =01 = 10 , 
&c.. 
then we have identically. 
00, 
01, 
02 
=K 
X , 
y , ^ 
10, 
11, 
12 
x' , 
y, 
20, 
21, 
22 
of'. 
f, z" 
and if the determinant on the right hand vanishes, that is if (x, y, z), {x', y', z'), (x", y", z") 
are points in a hne, then we have 
00 , 
01 , 
02 
10 , 
11 , 
12 
20 , 
21 , 
22 
