274 
MATHEMATICS: B. I. MILLER 
It is a great pleasure to express my sense of indebtedness to Dr. E. 
Newton Harvey, Prof. George Augustus Hulett, and Dr. Stewart Paton 
of Princeton University for timely advice and aid. 
A NEW CANONICAL FORM OF THE ELLIPTIC INTEGRAL 
By Bessie I. Miller 
The elliptic norm curve En in space Sn-i admits a group Gin^ of col- 
lin cations, and in fact there is a single infinity of such curves which ad- 
mit the same group. A particular En of the family is distinguished by 
a value of the parameter r, itself an elliptic modular function defined 
by the modular group congruent to identity (mod n). 
In the group G^n' there are certain involutory collineations with two 
fixed spaces. If En is projected from one fixed space upon the other, a 
family of rational curves Rm mapping the family of EnS, is obtained. 
The quadratic irrationality separating involutory points on En involves 
the modulus r and the parameter t of the Rm. When the genus of the 
modular group is zero and ;^ = 3, 4, 5, the irrationality can be used to 
define the elliptic parameter 
where a\ is the tetrahedral, octahedral, or icosahedral form. This is in 
contrast to Klein's form^ as developed by Bianchi,^ for there the normal 
elliptic integral is a rational curvilinear integral along an elHptic curve. 
A comparison of the two integrals is more illuminating if it is carried 
out for a special case. Let En be £5 in -5*4. In Bianchi's notation the 
five quadrics having £5 as their common intersection are 
(Pi '.ax}-\- Xi+2 - = 0, = x^, = 0, • • • 4), 
where a is the modulus. If a transformation of coordinates is made in 
order to bring into evidence the fixed spaces of the involutory collinea- 
tion used in the projection, then the icosahedral form which appears in 
the irrationahty is 
The integral Ui involving r = a expHcitly in a rather simple form is 
uniquely defined. Moreover it is invariant under all cogredient trans- 
DEPARTMENT OF MATHEMATICS. JOHNS HOPKINS UNIVERSITY 
Presented to the Academy. March 25. 1915 
