The great importance of the group of the equation in connexion with the nature of its integrals cannot here be dealt with, but it may be pointed out that if all the integrals of the equation are algebraic functions, the group must be a group of finite order, since the set of quantities y1, y2 ..., y_n can then only take a finite number of distinct values. _Groups of Finite Order._ We shall now pass on to groups of finite order. It is clear that here we must have to do with many properties which have no direct analogues in the theory of continuous groups or in that of discontinuous groups in general; those properties, namely, which depend on the fact that the number of distinct operations in the group is finite. Let S1, S2, S3, ..., S_N denote the operations of a group G of finite order N, S1 being the identical operation. The tableau S1, S2, S3, ..., S_N, S1S2, S2S2, S3S3, ..., S_NS2, S1S3, S2S3, S3S3, ..., S_NS3, . . . . . S1S_N, S2S_N, S3S_N, ..., S_NS_N, when in it each compound symbol S_pS_q is replaced by the single symbol S_r that is equivalent to it, is called the multiplication table of the group. It indicates directly the result of multiplying together in an assigned sequence any number of operations of the group. In each line (and in each column) of the tableau every operation of the group occurs just once. If the letters in the tableau are regarded as mere symbols, the operation of replacing each symbol in the first line by the symbol which stands under it in the pth line is a permutation performed on the set of N symbols. Thus to the N lines of the tableau there corresponds a set of N permutations performed on the N symbols, which includes the identical permutation that leaves each unchanged. Moreover, if S_pS_q = S_r, then the result of carrying out in succession the permutations which correspond to the pth and qth lines gives the permutation which corresponds to the rth line. Hence the set of permutations constitutes a group which is simply isomorphic with the given group. Every group of finite order N can therefore be represented in concrete form as a transitive group of permutations on N symbols. Properties of a group which depend on the order.