664 Wisconsin Academy of Sciences , Arts, and Letters. 
along the cuts ai of a surface T defined by F n =o and of defi¬ 
ciency p. We will put for the v 9 s the normal integrals dimin¬ 
ished each by a constant e. We shall then have 
O'v i, v,. 
V 
= 02 
0 C 
Zh \ 
1 J 
<p 
dUr 
— e, 
where the s and c’s are arbitrary points and <p’s are variable. 
These integrals of the first kind exist in the original surface 
Fn and therefore this 0 function is like branched with the sur¬ 
face. If it does not vanish identically we know from the prop • 
erties of 0 functions that it has p zeros on the surface. If how¬ 
ever, this $ function does vanish identically one or more of these 
zeros become arbitrary. 
Suppose that the e’s are so chosen that #is different from zero. 
Then 
i=P rx t 
' 6 &= 2 I 
i — 1 £ 
where x t are zero points of the 0 function and k h is independ- 
pendent of e tl . If moreover, the e’s are so chosen that 
0 («„ e 1 ,....e p )=o, 
then 0(»i, . v p )=o, 
and for an arbitrary point and we put 
where the point systems x i and x\ belong to an equation <$> = o 
besides lying on the surface F n =o. The quotient of the pro¬ 
duct 
(iu 
and the product 
