38 
THEOREM RELATING TO THE FOUR CONICS WHICH TOUCH THE 
[390 
From the last equation we have 
(cq - a 4 ) = 2 {0 - a V (X) - a' V (X')} - 2« 4 {d> - V (X) - V (X')} 
= 2 (a 4 — a 4 ) <I> — 2 (a - a 4 ) V (X) — 2 (a' — a 4 ) V (X'); 
that is 
( a i - a 4 ) $ - 2 (a - a 4 ) y № -2(a'~ a 4 ) V (X') = 0; 
or substituting for V (X), V (X 7 ) their values in terms of <E>, we find 
(a - a 4 ) (a 4 - a,) (a' - a 4 ) (a 4 - a 3 ) 
a, — a. — 
a — a., 
a -a 3 
= 0, 
which may be written 
(a ' - 1 *> i 1 +Hi) - <* - a *> ( : + Hf) - °- 
that is 
or again 
that is 
or finally 
(a 2 —a 4 )(a 2 —a 4 ) (a 3 — a 4 ) (a 3 - a 4 ) _ 
a, + a., - a 4 - a 4 + ^ = 0 ; 
a — a, a — a, 
(o,! _ a,) (* + Hi) + ^ _ a4> i 1+ Hi) = °’ 
, .a — a 4 , .a — a. 
(«2 - ai) (a - a 4 ) (a' - a 3 ) + (a 3 - a 4 ) (a - a 2 ) (a' - a 4 ) = 0, 
which is a known form of the relation 
1, 
a + a', 
aa 
=0, 
1, 
«1 + «4, 
a,a 4 
1, 
®2+ a 3> 
a„a s 
which 
gives the involution of the quantities a 
a; 
«i» «45 
We have in like manner 
1, 
ft -f Cl , 
a'a" 
= 0, 
1, 
a, + a,, 
a i a 2 
1, 
a s + a 4 , 
a 3 a 4 
and 
1, 
a" + a, 
a"a 
= 0, 
1, 
«i + <* 3 , 
«1*3 
1, 
a 2 + a 4 , 
a 2 a 4 
which 
give the involutions of 
the systems a, 
a" ; 
*1, a 2 ; 
respectively.