are the finite equations of a continuous group, and if C with parameters c1, c2, ..., c_r is the operation which results from carrying out A and B with corresponding parameters in succession, then the c's are determined uniquely by the a's and the b's. If the c's are rational functions of the a's and b's, and if the a's and b's are arbitrary rational numbers of a given corpus (see NUMBER), the c's will be rational numbers of the same corpus. If the c's are rational integral functions of the a's and b's, and the latter are arbitrarily chosen integers of a corpus, then the c's are integers of the same corpus. Hence in the first case the above equations, when the a's are limited to be rational numbers of a given corpus, will define a discontinuous group; and in the second case they will define such a group when the a's are further limited to be integers of the corpus. A most important class of discontinuous groups are those that arise in this way from the general linear continuous group in a given set of variables. For n variables the finite equations of this continuous group are x'_s = a_(s1)x1 + a_(s2)x2 + ... + a_(sn)x_n, (s = 1, 2, ..., n), where the determinant of the a's must not be zero. In this case the c's are clearly integral lineo-linear functions of the a's and b's. Moreover, the determinant of the c's is the product of the determinant of the a's and the determinant of the b's. Hence equations (ii.), where the parameters are restricted to be integers of a given corpus, define a discontinuous group; and if the determinant of the coefficients is limited to the value unity, they define a discontinuous group which is a (self-conjugate) subgroup of the previous one. The simplest case which thus presents itself is that in which there are two variables while the coefficients are rational integers. This is the group defined by the equations x' = ax + by, \ > y' = cx + dy, / where a, b, c, d are integers such that ad - bc = 1. To every operation of this group there corresponds an operation of the set defined by az + b z' = ------, cz + d in such a way that to the product of two operations of the group there corresponds the product of the two analogous operations of the set. The operations of the set (iv.), where ad - bc = 1, therefore constitute a group which is isomorphic with the previous group. The isomorphism is multiple, since to a single operation of the second set there correspond the two operations of the first for which a, b, c, d and -a, -b, -c, -d are parameters. These two groups, which are of fundamental importance in the theory of quadratic forms and in the theory of modular functions, have been the object of very many investigations. Discontinuous groups arising from geometrical operations.