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[523
523.
ON THE TRANSFORMATION OF UNICURSAL SURFACES.
[From the Mathematische Annalen, vol. in. (1871), pp. 469—474.]
I consider the question of the transformation (Abbildung auf einer Ebene) of
unicursal surfaces. Taking (x, y, z, w) for the coordinates of a point on the surface,
(x', y', z') for those of the corresponding point on the plane; then if X', Y\ Z', W'
denote each of them a function {oc, y', z) n ’, the equations of transformation are
x : y : z : w = X' : Y' : Z' : W:
and assuming that each of the curves
X'=0, Y' = 0, Z' = 0, W' — 0
(or, what is the same thing, the general curve
aX' + bY' + cZ' + dW' = 0)
passes once through each of ^ points, twice through each of a 2 points, ... , r times
through each of a r points (for convenience I write a r instead of a/); and writing also
n=n' 2 — tr 2 a r ,
0 = \n (n' + 3) — 3 — © — (r + 1) a,.,
(where © is = 0 or positive except in the case of special relations between the positions
of the fixed points a l , a 2 , ...,a r ), which equations give
— n = Sn — 6 — 2© — 2ra r ;
then the order of the surface is = n, and the order of the nodal curve is b=^(n—2)(n—3)+©.
I assume that the nodal curve has h apparent double points and t actual triple points,
but no stationary points, so that q being the class, we have q = b 2 — b — 2h—6t’, and
I endeavour to find these numbers q, t, h.
