Pengra \— Conformal Representation of Plane Curves. 069 
means of which the quadric can be expressed as a cone in four 
different ways. Our normal curve, the twisted sextic, must lie 
on both quadric surfaces as well as on the cubic surface. 
Hence it degenerates into a twisted cubic twice repeated. The 
two cones being evidently of the form 
<j) f <f> t — cj) 
</> 4 ( F x (</>’ s) ]—[ F 2 i 4 >’ 8 ) ] :3 =o 
containing a straight line in common. This gives the hyper- 
elliptic case for p= 4. 
For the case p=5 we know that there exist three quadratic 
relations among the <f>’ s. Let one of these give a cone in space 
of four dimensions. To the infinite number of curves which 
are tangent to the original curve at one point correspond the in¬ 
finite number of three flats tangent to the cone, and to the 
point of tangency correspond the four points of the normal 
curve found on each of the planes along which the three flats 
are tangent to the cone. The normal curve, which is a twisted 
curve of eighth order in space of four dimensions, is character¬ 
ized in this way by a G\ of points lying on a plane. 
If two of the three quadrics are cones the normal has two 
Gl’s and for three cones three Gl’s. 
If four quadratic relations exist the case is hyperelliptic. 
In conclusion, I wish to acknowledge the kindness of all the 
Professors of Mathematics at Wisconsin, and particularly my 
indebtedness to Dr. Dowling for his valuable suggestions and 
assistance in connection with the preparation of this paper. 
University of Wisconsin, July 14, 1903. 
