37 
390] SAME TWO LINES AND PASS THROUGH THE SAME FOUR POINTS. 
and by attributing the signs + and — to the radicals, we have, corresponding to the 
four conics, the equations 
(a - a,) V (X) + (a' - a,) V (X') + (a" - a,) V (X") = 0, 
- (a - a0 V (X) + (a' - a,) V (X') + (a" - a 2 ) V (X") = 0, 
(a - a 3 ) V (X) - (a' - a 3 ) V (X') + (a" - a 3 ) V (X") = 0, 
(a - a 4 ) V (X) + (a' - a 4 ) V (X') - (a" - a 4 ) V (X") = 0, 
where cc J} a 2 , a 3 , a 4 are the values of a for the four conics respectively. 
Eliminating a" we obtain the system of three equations 
(2a -a,- a 2 ) V (X) + (a 2 - «0 V (^') + («2 - «0 V (X") = 0, 
(«3 - «0 V № + (2a' - «> - a 3 ) V (X') + (a 3 - «0 n/ (X") = 0, 
(«! + ot 2 - a 3 - a 4 ) V (X) + (otj + a 3 - a, - a 4 ) V (X') + (a 4 + a 4 - a. 2 - a 3 ) V (X") = 0, 
and then eliminating the radicals we have 
2a — a 4 — a 2 , a 2 — a 4 , a 2 — a, 
a,-«! , 2a'-a 1 -a 3 , a 3 - a, 
a 4 + a, - a 3 - a 4 , a 4 + a 3 - a 2 - a 4 , a 4 + a 4 - a, — a 3 
= 0, 
which is in fact 
1, 
1, 
a +a', aa' 
a i + a 4, «I a 4 
a 2 + a 3 , a 2 a 3 
as may be verified by actual expansion; the transformation of the determinant is a 
peculiar one. 
The foregoing result was originally obtained as follows, viz. writing for a moment 
a V (X) + a'V (X') + a" V (■X") = ©, 
V(X)+ V(X')+ V(X") = 0, 
the four equations are 
0 - a, = 0, 
0 — a 2 = 2 (a — a 2 ) V (X ), 
© — a 3 <£» = 2 (a' -a 3 )V(^'), 
0 _ a 4 cp = 2 (a" - a 4 ) V (X") ; 
these give 
(a 4 - a,) <ï> = 2 (a - a 2 ) \J (X ), 
(a 4 - a 3 ) <î> = 2 (a' - a 3 ) V (X' ), 
(a 4 - a 4 ) = 2 (a" - a 4 ) V (^")-