668 Wisconsin Academy of Sciences, Arts, and Letters. 
identically the normal curve is such that it has an infinite 
number of tangent <t> curves. Any curve then of the same de¬ 
ficiency, in case the even theta function vanishes identically, 
has an infinite number of tangent curves. In particular the 
sextic of deficiency 3 can be put into the form 
(A 2 <£f2A4> 3 -M> 3 ) 
thus showing the sextic as the envelope of a quadratic sheaf 
of adjoint curves of order three. Conversely if we get any sort 
of a quadratic relation among the </>’s in the case p=3 an even 
theta function most vanish identically, for we can construct a 
system of such Abelian functions by putting the conic in the form 
L x L 2 —LI -o and Weber has proved* that if such a system can be 
constructed linearly and homogeneously from m independent 
Abelian functions then there can be found a characteristic w 
possessing the property that the function 0 [w] together with 
all its derivatives up to and including those of order m—1 must 
vanish identically. 
In case then that one quadratic relation exists among the 
<£s for p =3 the normal curve is a doubly covered conic and the 
case is hyper elliptic. 
For p=4 suppose that one even 0 function vanishes identi¬ 
cally. We then get the relation 
For a general curve of deficiency four the normal curve is, as 
we know, a twisted sextic made by the intersection of a quad¬ 
ric surface with a cubic surface. Now with the vanishing of 
the 0 functions the quadric surface becomes a cone and the 
representation is characterized by the fact that the tangent 
<t> curves to the original curve correspond to the planes 
tangent to the cone, and so to the points of tangency of </> curves 
correspond three points on the generator of a cone. The nor¬ 
mal curve is a twisted sextic such that it has a G'J or a singly 
infinite system of points three in a set, such that each group of 
three lies in a straight line. 
Suppose another quadratic relation to exist among the <f> s. 
This means • that the quadratic relation as determined to 
represent the surface F n contains a variable parameter by 
* Weber in Vol. XIII. Math. Ann., pp. 34-38. 
