184 
ON BICURSAL CURVES. 
[510 
that is 
VQ = i 
c-( 1 - b) 2 
— c 
and the corresponding value of \JQ' is 
where q stands for its value 
We may write these in the form 
1 : \<JQ : WQ' = Md : 1 4- d 2 : V®, 
where M is a constant, and ® is a quartic function of d, such that (1 4- 6 2 ) 2 — © is 
a quadric function only of 9. 
The equations 
0, y, *) = (1, u) m ( 1, v) n 
thus assume the form 
(#, y, z) = (Md, 1 + 8- + \/©) m {Md, 1 4- d 2 — V©) n 5 
and on the right-hand side the term of the highest order in d is 
(1 +9 2 + V©) m (1 4- 6> 2 - v©) n , 
viz. if n — or > m, then this is 
This is 
{(1 + d 2 ) 2 - ©} m (1 + d 2 - v©) n-m . 
= (1, d) 2m (1 4- d 2 - V©) n_m , 
which is of the order 2m 4-2 {n — m), =2n (which, in virtue of n = or > m, is = or > m + n). 
In particular, if n = m, then the highest order is =2n; or the curve of the order 2n, 
as represented by the equations 
{x, y, z) = (1, u) n (1, v) n , 
where (u, v) are connected by a quadriquadric equation, is also represented by the 
equations 
(x, y, z) = ( 1, d) n 4- (1, 
which is the required transformation of the original equations. 
It is to be noticed that the foregoing form, 
1 + u 1 
4- 2buv 
4- cv?v 2 = 0,