Pengra-—Conformal Representmtion of Plane Curves . 659 
constants is m+ 1, which is the number which the Riemann- 
Roch Theorem would give. Similarly for p — 1 one <f> function 
exists and one equation of the set (III) T = o’and the Riemann- 
Roch Theorem holds here also. 
To take up the language of the analytic geometry, we have 
selected a complete set of linearly independent functions each 
of weight m belonging to F n and we use these as co-ordinates, 
this fixing some sort of curve. Every point of F n gives rise to 
a set of values of these functions, or to a point, hence the whole 
surface F n may be conformally represented by the points of 
some curve. The functions which we select for this purpose 
are the Fs of which there are p linearly independent. 
Their ratios are functions belonging to the surface for they 
have only a finite number of infinities and these algebraic, and 
the ratios have one and only one value corresponding to each 
4>i 
point of F n . For if one of them, say^ assumed the same 
value for two different points of Fn, we should have a relation 
existing among the coefficients of and Fi, but by hypothesis 
the Fs are independent and hence their ratios belong to the 
surfaced,,. Since, as before shown, each becomes zero in 
2 p — 2 variable points of Fn, each ratio may become °o in 
2 p — 2 points, and zero in as many more, and hence the func¬ 
tions which are ratios of the Fs are of weight 2 p — 2. 
Although the Fs are linearly independent certain relations 
of higher order exist among them. For the case p = o, no <f> 
function exists and F„, as we saw, is representable by the points 
of a plane singly covered, or if we consider only the real points, 
by the points of a straight line. 
For p = 1 one function exists. We found that the simplest 
function which will represent Fn in this case is a function 
P (u | wi w> ) of weight two. According to the Riemann-Roch 
Theorem there are m — p + t 4-1 or two homogeneous arbi¬ 
trary constants in our representative function. Hence, since 
it is represented on a two-leaved Riemann’s surface, in the lan¬ 
guage of curves the simplest representation of F (x, y) — o is a 
doubly covered straight line. 
For p =2 two <f> curves exist. Our normal curve then is a 
doubly covered straight line since it exists in space of one 
dimension and the ratio of the Fs is a two valued function. 
No relation can exist between the Fs. 
