which are inverse to each other with respect to a given circle, x' and y' are rational functions of x and y, and conversely. Thus the group may be presented in a form in which each operation gives a birational transformation of two variables. If x + iy = z, x' + iy' = z', and if x', y' is the point to which x, y is transformed by any even number of inversions, then z' and z are connected by a linear relation z' = ([alpha]z + [beta])/([gamma]z + [delta]), where [alpha], [beta], [gamma], [delta] are constants (in general complex) depending on the circles at which the inversions are taken. Hence the group may be presented in the form of a group of linear transformations of a single variable generated by the two linear transformations z' = ([alpha]1z + [beta]1)/([gamma]1z + [delta]1), z' = ([alpha]2z + [beta]2)/([gamma]2z + [delta]2), which correspond to pairs of inversions at AB, BC and BC, CA respectively. In particular, if the sides of the triangle are taken to be x = 0, x^2 + y^2 -1 = 0, x^2 + y^2 + 2x = 0, the generating operations are found to be z' = z + 1, z' = -z^(-1); and the group is that consisting of all transformations of the form z' = (az + b)/(cz + d), where ad - bc = 1, a, b, c, d being integers. This is the group already mentioned which underlies the theory of the elliptic modular functions; a modular function being a function of z which is invariant for some subgroup of finite index of the group in question. The triangle ABC from which the above geometrical construction started may be replaced by a polygon whose sides are circles. If each angle is a submultiple of two right angles or zero, the construction is still effective to give a set of non-overlapping regions, which represent graphically the group which arises from pairs of inversions in the sides of the polygon. In their analytical form, as groups of linear transformations of a single variable, the groups are those on which the theory of automorphic functions depends. A similar construction in space, the polygons bounded by circular arcs being replaced by polyhedra bounded by spherical faces, has been used by F. Klein and Fricke to give a geometrical representation for groups which are improperly discontinuous when represented as groups of the plane. Group of a linear differential equation. The special classes of discontinuous groups that have been dealt with in the previous paragraphs arise directly from geometrical considerations. As a final example we shall refer briefly to a class of groups whose origin is essentially analytical. Let d^_ny d^(n-1)y dy ----- + P1 -------- + ... + P_(n-1) -- + P_ny = 0 dx^_n dx^(n-1) dx