666 Wisconsin Academy of Sciences, Arts, and Letters. 
The functions 
will not vanish identically for the points Xi, % taken arbitrar¬ 
ily, and therefore each of them will vanish in p —1 points beside 
e and both together in 2 p — 2 points on a function <f>. Now 
among the zeros of the first functions are xi, X2 . x m -i 
and let the rest be y,n, ym+ i . yo- i and let the zeros of the 
second one be “i, a 2 , a 3> ... a m _ h Pm+i, . ftp- i. There 
exists then a function <t> with the zeros xi , X 2 ,_ x m —h Vm, 
2/ra-H, Up— 1, a l» a 2. • • • - a m— 1, 1, .... ftp— 1. 
2/ra-H, Up— a l> a 2. • • • - a m— 1, 1, .... fip—1. 
Moreover since the zeros must satisfy the congruences 
Combining the last two by substraction we obtain 
dun —0 
It follows from Abel's theorem that there exists a rational 
function <f> which is infinitely small of the first order at the 
