ALGEBRAIC FUNCTION'S WITH AIJTOMORPHIC FUNCTIONS. 
13 
Let t=u^iv, if we measure the distance between two points, in the non- 
r I dt I 
Euclidian sense, by I taken along the circle orthogonal to the real axis and joining 
the points, then we can easily prove that the lengths of corresponding sides of the 
polygons are in this sense all equal; if we measure the area of any region by 
taken over that region, we can show that the areas of all the polygons are also in 
this sense equal; and the areas and lengths of corresponding regions and lines in the 
polygons are all equal. The substitutions by ivhich the polygons ai'e derived from 
each other are, in this non-Eiiclidian sense, simple displacements, which leave their 
dimensions unchanged. All the theorems of Lobatchewski’s geometry hold if, where 
Lobatchewski uses the word “ straight line,” we understand “ circle orthogonal to 
the real axis.” 
Thus, in non-Euclidian phraseology, we can say that the network of polygons has 
been obtained by drawing a rectilinear polygon of (n + 1) sides, deriving new poly¬ 
gons from it by turning the polygon through an angle tt round the middle points of 
its sides, and deriving fresh polygons from these by the same process, until the whole 
non-Euclidian plane is covered. This enables us to see that our figure is the naturcd 
extension of the division of a ivhole plane into parallelograms, so familiar in the theory 
of elliptic functions. For that division can be obtained by drawing any rectilinear 
triangle in the Euclidian plane, deriving fresh triangles by turning it through an 
angle tt round the middle points of its sides, and deriving new triangles from 
these by the same process, until the whole Euclidian plane is covered. The groups 
for which the elliptic functions are automorphic are sub-groups of the groups so 
obtained ; and similarly the groups, whose automorphic functions are required in the 
uniformisation of algebraic forms of genus higher than unity, are sub-groups of the 
group we have found. The reason why we have to pass from Euclidian to non- 
Euclidian geometry is, that in the Euclidian plane it is impossible to obtain a recti¬ 
linear figure with more sides than three, the sum of whose angles is tt. 
If to the original polygon we apply the substitution S.„+i, the point is 
unchanged, and the arcs D,j+2D„+i and D„+iC,i are transformed into each other. So 
the parts of the boundary of the polygon which correspond to each other in the 
transformations of the group are D„+2D,j+i to C,jD„+i, C^D,, to . . ., CiDj to 
E,i+2Di> respectively. If now we suppose the polygon lifted up from the plane, and 
these corresjjonding arcs pieced together, we obtain a simple closed surface, without 
multiple connectivity. 
Therefore the genus (genre, Geschlecht) of the group (as defined by Poincare) is 
zero. The group however may have, a,nd will in fact be proved to have, sub¬ 
groups whose genua is greater than zero. 
