Pengra—Conformal Representation of Plane Curves. 667 
points a* Pi and infinitely great of the first order at the points 
Xi yi and otherwise continuous and different from zero. This 
function is expressible as the quotient of two functions <£. 
But since there is a function <f> which vanishes in all the 2p—2 
0 _ 
points we have the two functions t <£=<£[ and ^ — </>* the first 
infinitely small of the second order in the points 
^11 ^3, ^3. Xm-v y m +n . yp-U 
the second infinitely small of the second order in the points. 
<x x , (X%, (X s ,.... e •»»<Xrn —1, Pm, 1, 2,. fip—1. 
These functions are then squares of Abelian functions. Elimi¬ 
nating r we obtain Vp p = <t> or 4>\ Pi = <A 2 . 
If we choose the points xi otherwise we obtain an arbitrary 
number of Abelian functions V</>i, V<£ 2 , etc., which have the 
property that the square root of the product of two of them is 
again an Abelian function. Since m—1 zeros are arbitrary 
we get m linearly independent Ps of this sort. 
I wish now to examine some special cases which arise here 
in the vanishing of the 0 functions. For surfaces of deficiency 
0, 1 or 2 no such relations can exist. For the case p=S there 
are three linearly independent P s. Ordinarily as we have seen 
no relation exists among them of lower degree than the fourth. 
The normal curve in the general case is a quartic which is 
fixed when we stipulate that it shall be a function belonging 
to the surface defined by the origir al equation, and fix its in¬ 
finities. 
If now an even 0 function belonging to the surface vanishes 
identically we get, as we have seen, the relation 
Pi 4> > — P=0 
We may regard this as the formal curve and, for the sake of 
continuity, say that it is doubly covered, thus our quartic rela¬ 
tion degenerates into two identical equations of the second degree 
and the normal curve is a conic doubly covered. This equation 
may be put into the form 
(A 2 <£ 2 +</>) (/< 2 <£ 1 + 2 /o£ 2+0)—[A// < ^ 1 +(A-f-//)^ > 3 +</ , l 2 -o 
thus showing up the tangent lines if we regard the Ps as co¬ 
ordinates, In the case then that an even 0 function vanishes 
