Colloquium publications.

226

THE MADISON COLLOQUIUM.

function
Q(t, 1)... Q(t, T,),(t) Q= (t, T1).. * (t, Tn)'
This function has the desired zeros and poles. Moreover,
(ta) = <(t),
2 2 w a~'I k,1(tpS+) = e k'a (t) = (t)
where a = 1, * *, p, and this completes the proof.
~ 13. PARAMETRIC REPRESENTATION OF A HOMOGENEOUS
ALGEBRAIC CONFIGURATION
Let
(1) F(w, z) = 0
be an irreducible algebraic equation. Let homogeneous variables
be introduced. If, for example, we wish to regard (1) as a curve
in the projective plane, we shall set
X2 X3
(A) z =-, w -.
On the other hand, we may regard (1) as a curve in the plane
of analysis, and set*
Z2 )2
(B) z = -, w -.
21 W1
Let (1) be uniformized as above by the function
z = p(t).
Then we can express z, w in terms of t by the aid of the prime
function as follows:
n(t, bl)... * Q(t b,), W(t, 1)... (t, OM)
( 2)(t, a,).* *, ) (t, ai)... (t, a (t, am)
Here, the points of ~ equivalent to al, -* *, a, are distinct from
those of A equivalent to bl, * *, b,; and similarly for ai, * *, am
and fl, *, Om.
* Cf. Riemann, Abelsche Funktionen, ~ 6; Werke, 1. ed., p. 103.