A New Canonical Form of the Elliptic Integral.

The elliptic norm curve Qn in space S„_i admits a group G2ni of collineations, and in fact there is a single infinity of such curves which admit the same group. A particular Qn of the family is distinguished by a value of the parameter t , itself an elliptic modular function defined by the modular group congruent to identity (mod. n). In the group G2n2, there are certain involutory collineations with two fixed spaces. If Qn is projected from one fixed space upon the other, a family of rational curves Rm mapping the family of Q„'s is obtained. The quadratic irrationality separating involutory pairs on Qn involves the modulus t , and the parameter t of the Rm. When the genus of the modular group is zero and n = 3,4,5, the irrationality can be used to define the elliptic parameter