660 Wisconsin Academy of Sciences, Arts, and Letters. 
For p = 3 one quartic relation exists among the <£’s so that 
our normal curve is a plain quartic. For we know from 
the Riemann-Roch Theorem that the most general quartic 
relation among the <£’s contains 4 (2 p — 2) —p-f-r -f 1= 14 ar¬ 
bitrary constants which are just enough to give the most general 
quartic relation among the <Fs since if we write out a quartic 
relation it will contain fifteen terms and by a selection of the 
fourteen arbitrary constants the function is completely fixed. 
For p=4 two relations of higher order exist, one of the second 
degree and one of the third degree. This may be proved as 
follows: We write out all the homogeneous functions of the 
second degree obtained by taking the squares of the different 4> ? s 
and their products taken two at a time. Divide each of these by 
some homogeneous function of the second degree in the <£’s. 
According to Riemann such functions like branched on a surface 
T can be expressed linearly in terms of 3 p —2 of them which 
make 3p — 3 linearly independent ones. There are —%—■ 
different combinations of the <£’s mentioned above and these can 
be expressed in terms of 3 p —3 independent ones, so there 
must exist at least M3p-3> or - quadratic 
relations among the <f>s. Similarly there are P ^ ^ -- ) 
combinations of the <£’s of the third degree. Divide each of 
these combinations by the same cubic relation among the <A’s. 
of these quotients 5 (p — 1) are independent of each other.* 
There must exist then at least 
p (p 4-1) (p 4-2) 
5 (p — 1) cubic 
relations among them. But we know from the preceding that 
there are at least 
(p —2) (p — 3) 
quadratic relations among the 
<£’s. Cubic relations among the <t >’s could consist of these quad¬ 
ratic relations multiplied by any one of p linearly independent 
equations of the first degree among the <£’s. To get the number 
ot cubic relations which do not break up thus we shall have to 
* Jahresbericht der Dautschen Matheuoatikor Ysrainigung, Vol- III, p. 
445. Math. Ann. Vol. XII, pp. 268-310. 
