AND IMAGTNABY BOOTS OF EQUATIONS. 
(79) We must now calculate J, D, L : 
c, —c, 0) 
~,Mc 6 (a 2 -cJ ; 
or writing 
D=», 
r 
D,=^ 
W- 
x’—(a?-\- c 2 )x 3 y 2 + a 2 c 2 xy 4 , 
jtf 2 0 — 
10 ’ ’ 5 
so that the coefficients of the biquadratic Emanant are 
(1, o,-^ 2 , 0, 0 \x, y y 
x; 
flj+C 2 
10 
ffl 2 + C 2 
a 2 c 2 « 2 c 2 
y\ -no -s-yj 
Hence the quadratic covariant becomes 
A 2 
5 
UT ^ 2 + ^(« 2 +c 2 )«W+ ihd(« 2 + c2 ) v 
20« 2 c 2 + 3 (a 2 4- c 2 ) 2 . , 2, „ . 2V 2 2 , 2 
= loo ^+25 (« 2 +c 2 )(aV)/. 
Hence, by definition, J (which = — 4 x Discriminant of the Quadratic Covariant) 
= - x4o (« V X« 2 + c 2 ) (3 (a 2 - c 2 ) 2 + 32 a V) ; 
and making 
T J 
J /-(a 2 + C 2 ) g ’ 
T 6 64 6 2 8 
J;— 625^ 625 ^ 5 4 ^ 5 5 ^" 
Finally, to calculate L. The canonizant of the form 
1 0 A 0 
0 A 0 B 
A 0 B 0 
3 / 3 ; —xy 2 ; a?y; — x 3 
is 
(A 3 — AB )x 3 + (B 2 — A 2 B)^?/ 2 , 
of which the discriminant is 
