662 Wisconsin Academy of Sciences, Arts, and Letters. 
projecting the twisted sextic from the vertex of the cone we get 
a conic three times repeated since each generator of the cone 
will cut the sextic in three points. This case is referred to 
later. 
Noether has published some work in Vol. 26 of the Mathe- 
matische Annalen in which he actually works out the rela¬ 
tions which may exist among the Fs for the cases p = 5, 6 and 
7. Kasbohrer has a dissertation on the case p — 8. 
If p — 5 our function of the second degree in the <f>’s contains 
fifteen homogeneous linear constants. According to the Rie- 
mann-Roch Theorem it should contain only 16 — 5 + 1 = 12 
When we fix these twelve there are left then three more, homo¬ 
geneous and linear. So we see the three linearly independent 
quadratic relations among the The normal curve in this 
case is a twisted curve of eighth degree in space of four dimen¬ 
sions. Weber proves* that if we take any three homogeneous 
functions of degree two in the <f>’s and eliminate two of the 
variables we shall get a curve of deficiency five, thus proving 
that any twisted curve of degree eight formed by the intersec¬ 
tion of three quadrics in space of four dimensions, represents a 
F (x, y) =o of deficiency five. 
It is very easy to get some properties of curves of higher or¬ 
der out of the properties of the normal curves. To illustrate 
this take the case p =3 where the normal curve is, as we know, 
a quartic with no double points. We will prove that a curve 
of order n and deficiency three may be regarded as the envelope 
of sixty-three different quadratic sheaves of curves of order 
2 (n — 3.) Six in each set break up into two curves each of 
order n —3 which pass through all of the double points of the 
curve of the nth degree and have their other intersection on a 
curve of degree 2 (n — 3). In particular a sextic of deficiency 
three possesses 28 tangent cubics which pass through the 
double- points of the sextic and such that they can be arranged 
into sixty-three sets of twelve each, such that the points of in¬ 
tersection of corresponding cubics in each set shall lie on a 
curve of order six having the same double points as the origi¬ 
nal sextic. For if we transform a given curve of deficiency 
three and order n by means of a net of adjoint curves of order 
n — 3 we get a quartic of defiency three and to the adjoints 
correspond straight lines. We know that such a quartic has 
* Math. Ann., Vol. XIII, p. 44. 
