154 
A Linear RELATION BETWEEN CERTAIN OF KLEIN’s X FuNcTIONS AND SIGMA 
Functions oF LowErR Division VALUE. By JoHn A. MILLER. 
Professor Felix Klein has defined a system of interesting functions by the 
following equation* : 
m— 1 
es (aya, EE a arora. ys (So. Net (1) 
m p—0 ing TY Seal ct 
m m 
Where « — 0, or 3, according as m is odd or even, and 
(77, +42, |) (u— 40, + now, ) 
Crepe a eal 2m Jo (UAW, + Meg | 01, @3)..(2) 
m 
u is the fundamental variable of the elliptic functions, ©,, », the periods of an 
elliptic integral of the first kind, 7,, 7, the periods of an elliptic integral of the 
second kind, Cy, a quantity independent of u and c(u|,,©,) is the ordinary 
Weierstrassian o-function and where /, and m are integers. 
I shall now prove that in the case m is a square number, 7. e., m —n? that 
ed (u) can be expressed as a linear homogeneous function of G (nu | @,, 2). 
he Lt 
n? eo 
n? 1 
To do this, we need the so-called Hermite Lawt which, when specialized to 
meet our needs, is as follows: 
Suppose we are given n quantities defined as follows : 
n 
m= Ci, I o(u—ai, 3) (Aa 
such that the sum of the zero points in the period parallelogram of the 1-plane is, 
= Xai, j = 0, 
And, suppose that we are given a o-product, 
20 - a Uw 
7? Il (wu— um) 
i=1 
Such that the sum of the vanishing points of Pin the period parallelogram of the u-plane is 
Pp flo)e™ + wns) ( wut 
n 
By 
i=1 
9 ! 
—— 70) 4 T He, 
*See Felix Klein: ‘‘ Vorlesungen iiber die Theorie der Elliptischen Modulfunctionen,’’ 
Zweiter Band, p. 261, equation 1. 
+F. Klein, Elliptische Normal-Curven und Modeln nter Stufe, p. 355, or Crelle’s Journal, 
Band XXXII, Hermites, Lettre i Mr. Jacobi. 
