ALGEBRAIC FUNCTIONS WITH AUTOMORPHIC FUNCTIONS. 
23 
Let + tSy and y,. — iS,^ be the double points of S,., 
then 
a, = y„ b,= — (yl + Sf.), a. 
Therefore 
1 . 
yi + Si 72 + §2 y\ + S3 
ri 72 73 
1 1 1 
= 0. 
This shows that the double points of all three substitutions lie on a circle 
orthogonal to the real axis. Since S2S3S4 is a self-inverse substitution, the double 
points of S4 lie on the same circle. 
Hence, if we attempt to construct the fundamental polygon, we find that all its 
angular points lie on the same circle orthogonal to the real axis, and therefore all its 
sides coalesce, and its area is zero. This explains why tlie method fails in this case. 
We now proceed to the uniformisation of algebraic forms ivhich are not hyper- 
elliptic. These only occur when the genus is greater than two. 
If we are given any algebraic form of genus ^0, it is known that it can by birational 
transformation be represented on a Liemann surface of which all the branch-points 
are simple, f.e., only two sheets interchange at any branch-point. 
Let f{u, 2) = 0 be an algebraic equation corresponding to this surface. Suppose 
the branch-points are at the values of z for which z = Cj, e-i, e^, . . . + respectively. 
It may of course happen that for some of these values of z there are several branch¬ 
points superposed on each other on the Riemann surface. 
Now in the z-plane, join the point c,j + 2 fo each of the points Cj, 62? • • • + and 
conformally represent this, in the plane of a variable t, on the fundamental polygon 
of a group from {n -j- 2) self-inverse substitutions, as before explained. 
Then, as before, z is a uniform function of t. At each of the points z = Cj, 63, . . . 
e,,+ 2, say e,., u is expansible in a series of ascending powers of either (z — e,.)" or 
(z — e,.), according as the point z = e., happens to be a branch-point or not in the 
sheet in which the point is situated. But near this point (z — ef is expansible in a 
power-series in terms of {t — to), where to is the value of t at the point; so in either 
case, u is expansible as a series of ascending powers of {t — to) ; that is, u has no 
branch-point, considered as a function of t, at this point. 
But since z is a uniform function of t, the only points where u can have branch¬ 
points, considered as a function of t, are the points where u has branch-points 
considered as a function of z; that is, the points Ci, e-i, . . . e,^ + 2. Hence, m is a 
uniform function of t. 
Thus, any algebraic curve can be uniformised by means of groups of substitutions 
formed from self-inverse substitutions. 
It will be seen that a great similarity exists between the place occupied by self¬ 
inverse substitutions, in the theory of groups of projective substitutions, and the 
