ME. A. CATLET ON THE ANALYTICAL THEOKT OF THE CONIC. 
641 
is the equation of the polar of the point (a’', y\ z') in regard to the conic, and that 
(A,..xr, r)=o 
is the line-equation of the pole of the line (|', K') 5 oi'? Avhat is the same thing, the 
point-coordinates of the pole are 
5. The inverse matrix is 
( 
h, 
9 
A, 
H, 
G ) 
h. 
f 
H, 
B, 
F 
9^ 
c 
G, 
F, 
C 
but it is convenient to disregard the factor — , and speak of (A, B, C, F, G, H) as the 
inverse or reciprocal coefficients. The equation just written down implies the relations 
A.a-{-13Lh-\-Gg=l^, A7«+H$+G/'=0, &c., which maybe arranged in two different ways 
as a system of nine equations. 
6. We have also 
(BC-F^ CA-G^ AB-H^ GH-AF, HF-BG, FG-CH)=K(a, c,/ A), 
and 
ABC-AF^-BG^-Cff+2FGH=K^ 
which are well-known theorems. 
7. I notice also the theorem 
(«, . .Jx, y, z)\ («, . .X^', y\ . .X^, y, zXx', y\ z')J 
= (A, ..\yz'—y’z, zx'—z'x, xy'—x'yf, 
which is much used in the sequel : it may be mentioned, in passing, that this is included 
in the more general theorem 
(a, . X^, y, zjl (a, . .Jx’, y\ z%l, m, n) 
{a , . .X^’, y, zjl’, m', Til), {a , . .X^', y', z'Jl, m, n) 
= (A, ..\yz!~y'z, zx'—z'x, xy’ —x'yymfil —m'n, Til'—n’l, 
which is at once deducible from 
L?+Mm+Nw, L7 +M'to+N'w 
L^'+M w^'+N7^', L7'+MW-j-NV 
by writing therein 
(L , M , N )={ax -^hy -^gz , hx -\-by +fz , gx Arfy -\rcz ), 
(L', M', N')=(a^'+%'+^^^ hx! -\-by' -\-fz' , gx' -^-fy' -\-cz'). 
4 s 2 
