658 Wisconsin Academy of Sciences, Arts, and Letters. 
L3t the p linearly independent integrals of the first kind of 
F a be wi, wi, ws, . Wp and their derivatives with respect 
to y be (f>L, <t>i, <£j,. 4>f,. It is easy to show that the 
Fs so found are linearly independent and that any p -f* 1 
can be expressed linearly in terms of the other p. By expand¬ 
ing w in the region of the zero points, and the branch points, 
and then differentiating to find the value of the <t>’s we deduce 
the fact that the Fs have 2 p — 2 variable zeros on F a and 
2 n zeros at the infinite points of F n . 
This work offers an easy proof of the Riemann-Roch Theo¬ 
rem, for the equations (II) become in terms of the <t>’s 
f Cl <Pi (y i) + c 8 (p 1 (y 2 ) + c 3 4> t (y a) 
I 
IIH 
I 0 2 icy A + 0 2 (y 2 ) + c 3 02 (y 3 
d -Cm, 0! ( ym ) = o 
. + C m 02 (ym) = O 
l c i <t>p (y\)-\-c 2 0 p (y 2 ) + c 3 <t>p (y 3) 
+ Crn<f>p (ym) — o 
since A, = — 2 7r i 
fi, 
dkj \ * 
ci y )y — yo 
If t of these equations are dependent upon the rest it is pos¬ 
sible to combine the other p — r in such a manner as to get 
these dependent ones, and indeed to get r equations which 
shall be linear in the Fs and which vanish in all the points 
Vu yi, ys, . y m , which proves that there are r linearly 
independent functions which vanish in all the points y 1 , yz, ys, 
. y>n. 
By solving the system (III) we can express p — rof the c’s in 
terms of the other m — p -f r. These m—p + t variables en¬ 
able us to fix the totality of algebraic functions belonging to 
Fa which are of weight m or less. The most general function 
of weight m belonging to Fa contains in general m—p + r + 1 
arbitrary constants. 
The Riemann-Roch Theorem so proved would hold only for 
p > 1. Klein extends it to the cases p = 0 and p = 1. He con¬ 
structs a function 
w — o. () 
Cm 
W — W,n 
which is evidently a function of weight m belonging to the sur¬ 
face. Since there are no Fs, r = 0 and the number of arbitrary 
* See Klein Theorie der Elliptischen Modulfunctionen, Vol. I., p. 532. 
