182 
ON BICURSAL CURVES. 
[510 
it is shown that such a transformation is possible, and a mode of effecting it, derived 
from a theorem of Hermite’s in relation to the Jacobian H, ® functions, is given in 
Clebsch’s Memoir “ Ueber diejenigen Curven, deren Coordinaten sich als elliptische 
Functionen eines Parameters darstellen lassen,” Grelle, t. lxiv. (1865), pp. 210—270. The 
demonstration is a very interesting one, and I reproduce it at the end of this paper. 
I remark, in passing, that the analogous reduction in the case of unicursal curves is 
self-evident; the equations 
{x, y, z) = ( 1, 6) n+k 
will represent a curve, not of the order n + k, but of the order n, provided there 
exist k common values of 6 for which the three functions vanish ; in fact, the three 
functions have then a common factor of the order k, and omitting this, the form is 
(x, y, z) = (1, d)\ 
Returning to the curves of deficiency 1, we see that a curve of the order 2m + 2n 
contains 6 (m+ n) constants, and is normally represented by a system of equations 
(x, y, z) = ( 1, d) m+n + (l, d) m+n ~ 2 ^/S. 
Such a curve may be otherwise represented : we may derive it by a rational trans 
formation from the curve (2) = 1) of the order 4 (binodal quartic), the equation of 
which is 
(1, uf (1, v) 2 = 0; 
viz. the coordinates are here connected by a quadriquadric equation ; and we then 
express x, y, z in terms of these by a system of equations 
(x, y, *) = (1, u) m { 1, v) n . 
It is, however, to be observed that the form of these functions is not determinate : 
each of them may be altered by adding to it a term {(1, u) m ~ 2 (1, v) n ~ 2 j {(1, u) 2 (1, w) 2 }, 
where the second factor is that belonging to the quadriquadric transformation, and 
the first factor is arbitrary. Using the arbitrary function to simplify the form, the 
real number of constants is reduced to (m + 1) {n + 1) — (m — 1) (n — 1), = 2(m + n); or 
the three functions contain together 6 (m + n) constants, one of which divides out. The 
quadriquadric equation, dividing out one constant, contains eight constants ; but reducing 
by linear transformations on u, v respectively, the number of constants is 8 — 6, = 2. 
Hence, in the system of equations, the whole number of constants is 6 (m + n) — 1 + 2, 
= 6 (m + n) +1 ; viz. this is greater by unity than the number of constants in the 
curve (D — 1) of the order 2 (m + n). The explanation of the excess is that the same 
curve of the order 2 (m + n) may be derived from the different quartic curves 
(1, u) 2 (1, v) 2 =0; this will be further examined. 
The transition from the one form to the other is not immediately obvious; in 
fact, if from the quadriquadric equation (1, uf (1, vf = 0 (say this is A + 2Bv -f- Cv 2 = 0, 
where A, B, C are quadric functions of u) we determine v ; this gives Gv = — B + VR 2 — AG,