holds between them. Now it may be shown that any continuous group of which X, Y are infinitesimal operations contains also (XY) among its infinitesimal operations. Hence if r linearly independent operations X1, X2, ..., X_r give rise to a finite continuous group of order r, the combinant of each pair must be expressible linearly in terms of the r operations themselves: that is, there must be a system of relations __k=r (X_iX_j) = \ c_(ijk)X_k, /__k=1 where the c's are constants. Moreover, from Jacobi's identity and the identity (XY) + (YX) = 0 it follows that the c's are subject to the relations c_(ijt) + c_(jit) = 0, \ | and > | [Sigma][s](c_(jks)c_(ist) + c_(kis)c_(jst) + c_(ijs)c_(kst)) = 0 / (iii.) for all values of i, j, k and t. Determination of the distinct types of continuous groups of a given order. The fundamental theorem of the theory of finite continuous groups is now that these conditions, which are necessary in order that X1, X2, ..., X_r may generate, as infinitesimal operations, a continuous group of order r, are also sufficient. For the proof of this fundamental theorem see Lie's works (cf. Lie-Engel, i. chap. 9; iii. chap. 25). If two continuous groups of order r are such that, for each, a set of linearly independent infinitesimal operations X1, X2, ..., X_r and Y1, Y2, ..., Y_r can be chosen, so that in the relations (X_iX_j) = [Sigma]c_(ijs)X_s, (Y_iY_j) = [Sigma]d_(ijs)Y_s, the constants c_(ijs) and d_(ijs) are the same for all values of i, j and s, the two groups are simply isomorphic, X_s and Y_s being corresponding infinitesimal operations. Two continuous groups of order r, whose infinitesimal operations obey the same system of equations (iii.), may be of very different _form_; for instance, the number of variables for the one may be different from that for the other. They are, however, said to be of the same _type_, in the sense that the laws according to which their operations combine are the same for both. The problem of determining all distinct types of groups of order r is then contained in the purely algebraical problem of finding all the systems of r^3 quantities c_(ijs) which satisfy the relations c_(ijt) + c_(ijt) = 0, [Sigma] [c_(ijs)c_(skt) + c_(jks)c_(sit) + c_(kis)c_(sjt)] = 0. s