Pengrd' — Conformal Representation of Plane Curves. 665 
may be put equal to —, <t >i and each having like branches 
<P 2 
(f>. 
with the original function. —- is a function of weight 2 p — 2 
<t>> 
since it is zero in 3 p — 2 variable points and in as many. 
The <f>’s are then adjoint curves of the original function and are 
of order n — 3. 
Introducing the 0 functions with characteristics we know 
that there are 2° -l (2'°—1) odd theta functions and 
2 v ~ 1 (2 P 4-1) even theta functions. In general only the odd 
theta functions vanish for the zero values of the arguments.* 
If we assume now that e h ~ — e h and f h =~ — f h we get the 
zeros of our function before considered to fall together in pairs 
and there exist p — 1 points at which the Ps are zero of order 
two. V<£ is an Abelian function and there are 2 P ~ 1 (2 P ± 1) 
of these together, one for each different characteristic. In the 
case of the odd theta functions there are 2 P ~ 1 (2 P —1) <£ curves 
tangent to the original curve. 
Let us make the assumption that up to any number m our 
function 
vanishes identically for all the points, xi t x* t . x m —\ and 
does not vanish. 
According to Riemann’s work before referred to, the condi¬ 
tion is that 
0 [w] (vj, — .... v p ) 
with all its derivatives up to and including the (m—l)s£ but 
not all the mth derivatives must vanish for the zero value of 
the arguments. 
*Ueber das Verschwioden der Theta FuDction.en, Riemann’s Werke, 
p.198. 
