158] 
A SIXTH MEMOIR UPON QUANTICS. 
581 
is the equation of a conic having double contact with the conic TJ — 0 at its points 
of intersection with the line P = 0. Such conic is said to be inscribed in the conic 
TJ = 0; the line P = 0 is the axis of inscription; this line has the same pole with 
respect to each of the two conics, and the pole is termed the centre of inscription: 
the relation of the two conics is completely expressed by saying that the four common 
ineunts coincide in pairs upon the axis of inscription, and that the four common 
tangents coincide in pairs through the centre of inscription; it is consequently a 
similar relation in regard to ineunts and tangents respectively; and it is to be inferred 
d priori, that if T = 0 be the line-equation of the conic U = 0, and II = 0 the line- 
equation of the centre of inscription, then the line-equation of the inscribed cone is 
T + /¿II 2 = 0. 
204. To verify this, I remark that if the equation of the axis of inscription be 
%x + rj'y + %'z = 0, 
then {ante, No. 201) we have for the line-equation of the centre of inscription 
n=(&,...$£ v> v, n-o. 
The line-equation of the inscribed conic is in the first instance obtained in the form 
(a-V, & + \(a,...#nS’- V % fy'-Pvy^O; 
but we have identically, 
= K(a,...$v?-v'S, h'-enY, 
and the equation thus becomes 
which is of the form in question. 
205. Take (x, y', z') as the point-coordinates of the centre of inscription, the 
equation of the axis of inscription is 
{a, b, c, f, g, hjoc, y, z\x, y, z) = 0; 
and we may, if we please, exhibit the equation of the inscribed conic in the form 
(a,...\x, y, z) 2 {a, y, z'Y o,o& 6 - {{a,.. .\x, y, z\x’, y', /)} 2 = 0, 
where 6 is a constant. This equation may also be written 
(a, ...Tjx, y, z) 2 (a,. ..\x, y, z') 2 sin 2 0\yz - yz, zx' - zx, xxj - x'y) 2 = 0, 
the two forms being equivalent in virtue of the identity, 
(a, y, z) 2 (a, y\ z') 2 -{{a, ...\x, y', z'\x, y, z)) 2 
= (E, • ~ y'z, zx' - z'x, xy' - xy) 2 .