1 
510] 
ON BICURSAL CURVES. 
183 
= — B + ^/D, suppose; and then substituting in the equations (x, y, z) = ( 1, u) m (1, v) n , 
we find 
{x, y, z) = (1, u) m (C, -B + Vil) n ; 
viz. we have (x, y, z) proportional to functions of u, involving the quartic radical \/Cl ; 
but these functions are of the order (not m + n, but) m + In. 
In particular, if n = m, then, instead of functions of the order 2n, we have functions 
of the order 3ft. The reduction in this last case to the form where the order is 2n 
can be effected without difficulty, but in the general case where m and n are unequal, 
I do not know how it is to be effected except by the general process explained in 
Clebsch’s Memoir. 
We may, in fact, by linear transformations on u, v, reduce the quadriquadric 
relation to 
1 + u 2 
+ 2buv 
+ v 2 + cu 2 v 2 = 0, 
that is 
1 + u 2 + 2 buv + v 2 + cu 2 v 2 = 0; 
or putting herein u + v = p, uv = q, the relation is 
that is 
1+p 2 — 2q + 2bq + cq 2 = 0, 
p 2 = — 1 + ( 2 — 26) q — cq 2 , 
jo 2 — = — 1 + (— 2 — 26) q — cq 2 ] 
viz. extracting the square roots, 
if for shortness 
u-\-v = \JQ, u — v = \J Q', 
Q = — 1 + ( 2 -2b) q — cq 2 , 
Q' = - 1 + (- 2 - 26) q - cq 2 ; 
we may then rationalise one of the radicals, for instance, Q ; viz. writing 
— c {— 1 + (2 — 26) q — cq 2 } = Icq — (1 — 6)} 2 + c — (1 — 6) 2 , 
then, if 
cq 
(i-6)=v c -(T-y.ifi)-b, 
this becomes