Pengm — Conformal Representation of Plane Curves. 657 
P (U I W,, Wo) = i + 2 ——— ---- 77 ,—;-- rz 
1 ” J u 2 (a — 7n 1 w 1 —m 2 w 2 ) i [m 1 w 1 -j- m 2 w 2 r 
P (u j w 19 u).>) = — 22 
{u — tn ± w x — in 3 w 2 ) 3 ‘ 
These two are everywhere finite except for w = o, where the 
former is infinite of the second order. Hence P (u | wi, wi ) is 
a function of weight two belonging to F n . By means of it we 
can represent F n conformally upon a two leaved Riemann’s 
surface. 
In case p > 1 we desire to build up a function of weight m 
which shall belong to the surface, by means of which the sur¬ 
face Fn may be conformally represented upon a simpler surface. 
Suppose that one such function of weight m exists on the sur¬ 
face and let it be represented by w and its m infinities by 
2/i, 2 / 2 , y-s ,.2 h%. Let these infinities be of the nature 
y—vi 
Let 
Vy t = h - v % h . j P 
where the v’s are the periods of w for the cuts ai and the fs are 
the normal integrals of the first kind, the periods of jk for the 
cuts ai being all zero except the period for au which is unity, 
and the periods for the cuts hi being Ki, 
■i y v — o 9 y v — c 3 y v 
y l -2 ■' : 
is everywhere finite, the possible infinities disappearing by sub¬ 
traction, and since it has periods for the 2 p cuts, it is an in¬ 
tegral of the first kind. Moreover, according to definition, the 
periods for the cuts ai are all zeros, and therefore this integral 
can be put equal to a constant,* and 
w ~ Q 0 + c x y y ^ + c 2 y y * + c 3 y y ^ + 
In order then for u to have but one value for each point on the 
surface, the periods across the cuts bt must be equal to zero and 
II 
cq A/fc 4- Cg X 2k + c » X Sk .+ C m X mk ~ °‘ 
If m > p + 1 the c’s can be found and F n can be conformally 
represented upon an m leaved Riemann’s surface spread over the 
w plane by means of a function of weight m belonging to F n . 
* See Klein Theorie der Elliptischen Modulfunctionen, Vol. I., p. £24. 
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