80 
ME. A. CAYLEY’S SIXTH IMEMOIE LTOX QUAXTICS, 
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 meunts comcide m 
pairs upon the axis of inscription, and that the four common tangents comcide m pairs 
through the centre of inscription ; it is consequently a similar relation m regard to 
ineunts and tangents respectively ; and it is to be inferred apnon, that if T=0 be the 
line-equation of the conic U= 0 , and n =0 the line-equatoof the centre of inscrip- 
tion, then the line-equation of the inscribed conic is 0 . 
204. To verify this, I remark that if the equation of the axis of inscription be 
then {ante. No, 201) we have for the line-equation of the centre of inscription 
The line-equation of the inscribed conic is in the first instance obtained in the form 
but we have identically, 
( 91 , . -Il, n, l )\% • Irl -In)- • 
and the equation thus becomes 
[K-fx(g, ..I?, t', f)’](a, -IS, 1. ?)'->■{ (a. -B'. 
which is of the form in question. 
205. Take as the point-coordinates of the centre of inscription, the equation 
of the axis of inscription is 
{a, b, c,f, g, hXx, y, z^x ', «/ , 2 ')= 0 - 
And we may, if we please, exhibit the equation of, the inscribed conic in the form 
(a, . . 1 ^, y. zf{a , . .Jjy, y\ 2 ')' cos^ . -fc y. zjj^, 3 /, 2 ')}^= 0 , 
where is a constant. This equation may also be written 
(«, . .1^, y. y'^ • -Xy^'-X^’ zcy-z'x, x^-3^yy=^. 
the two forms being equivalent in virtue of the identity, 
(a, . .\x,y,zr(a, . {(«, 
206. The line-coordinates (|', rj, I') of the axis of inscription are 
ax’-\-hy'-\-gz', Jixf +by' -\-fz' , g^-yfy'-\-cz', 
and we thence deduce the relation 
In order that the form 
(a, . .Jx, y, zf{a, . ■Ir', y\ z'f cos’ «-{(«,. •lx’, y', z'Jx, y,z)r=0 
may agree with the originally assumed form 
{a, . y, zf-\-'k{^x+ri'y-\-t!zy. 
