Pengra<-—Conformal Representation of Plane Curves. 663 
t wenty-eight bitangents which may be divided into sixty-three 
s ets of twelve each such that the points of intersection of cor¬ 
responding pairs lie on a conic. This quartic may be regarded 
as being the envelope of sixteen different quadratic sheaves of 
conics— each sheaf containing six conics which break up into 
two straight lines, forming a Steiner complex. Moreover all 
of these double points lie on the Jacobian of the net to which 
all of the sheaves belong, this Jacobian being of order six. The 
proposition then follows as the result of the correspondence be- 
t he curve and the nth degree and the quartic. 
Another illustration is here taken from the case p — 4. We 
will prove that there are twenty-seven different pairs of points 
o n a curve of degree n and deficiency four w r hich can be taken 
in sets of three in forty-five different ways to lie on as many 
adjoint <t> functions of the original function. 
For in this case the normal curve is the twisted sextic — the 
intersection of a quadric with a cubic surface. We know that 
through any straight line on a cubic surface can be passed 
five planes each of which cuts the surface in two or more lines, 
s o that each line is intersected by ten others — eight outside a 
plane containing three of them. 
Considering then three lines in a plane and the eight lines 
w hich cut each, we have twenty-seven lines in all. Each line 
i ntersects the quadric surface in two points thus giving two 
p oints of the normal curve. There are, therefore, fifty-four 
different points on the twisted sextic such that they lie by sixes 
in forty-five different planes since the twenty-seven lines lie 
by threes in forty-five different planes. Carried over by 
transformation to a curve of order n we get the proposition 
above. 
In the work hitherto, we have been considering the curves 
represented as perfectly general. We will now examine some 
special cases. In order to do this we take up some 0 functions. 
In 
6\v i, v 2 , Vg,.... vp) 
on 
= 
_QO 
Tt i(an 2 —(— 2 n v) 
let an 2 be a complete quadratic function of the n’s. and n a com¬ 
plete linear function of the v’s of which there are p. We will 
now put for the a’s the period moduli of the normal integrals 
* Salmon’s Geometry of Three Dimensions, p. 769. 
