ALGE 13 RAIC FUNCTIONS WITH AUTOMORPHIC FUNCTIONS. 
3 
tions of certain kinds of groups, which are described in § 3. These are either groups 
whose generating substitutions are of period two, or sub-groups of such groups, 
This theorem is made to depend on the well-known theorem that any algebraic form 
can, by birational transformation, be represented on a Riemann surface with only 
simple branch-j)oints. A method is given for the division of the Aplane into 
polygons, corresponding to a group generated by real substitutions of period two, 
whose double points are not on the real axis ; and the genus of the group is found. 
The group is of the kind called by Poincare Fuchsian; the polygons into which the 
plane is divided are simply-connected, and cover completely the half of the Aplane 
which is above the real axis. P^esults are deduced relating to the possibility of 
uiiiformising any algebraic functions by automorphic functions of such groups, and 
the analytical connexion of the uniformising variable with the variables of the form. 
In § 2, certain pi'operties of substitutions of period two are found, which are of use 
later. These substitutions are for brevity termed “self-inverse'' substitutions, owing 
to the fact that they are the same as their inverse substitutions. 
In § 3, a method is given for carrying out the division of the ])lane into polygons, 
corresponding to a group generated by a given set of self-inverse substitutions. It is 
proved that the genus of the group is zero, although the group has sub-groups whose 
genus is greater than zero. 
In § 4, the automorphic functions of the group are introduced. Since the group is 
of genus zero, these automorphic functions are all rational algebraic functions of one 
of them ; the conformal representation of the polygons in the Uplane on the plane of 
this variable is considered. It is shown that the functions which have been obtained 
solve the following problem of conformal representation :—-Draw from any point P) 
in the jilane of a variable 2 , lines (not necessarily straight) to any other points 
A, B, C. . . . This set of rays is to be regarded as the boundary of the s-plane, and 
the problem is, to conformally represent the z-plane, thus bounded, on a simply- 
connected region in the plane of a variable t, in such a way that each of the lines 
PA, PB, PC, . . . gives rise to two distinct lines of the boundary of the Uregion ; and 
one of these lines is derivable from the other by a projective substitution 
’ ct + dj ' 
The uniformisation of algebraic functions is afterwards made to depend on this 
problem of conformal representation. 
In § 5, the analytical relation between the variables « and t is discussed. It is 
shown that they are connected by a differential equation which is a particular case 
of what has been named by Klein the “ generalised Lames equation,” and has been 
connected by Bocher with the differential equations of harmonic analysis. 
In § 6, the functions which have been obtained are applied to the uniformising of 
algebraic forms. The differential equation in the hyperelliptic case is found to be the 
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