for all values of i, j, k and t. To two distinct solutions of the algebraical problem, however, two distinct types of group will not necessarily correspond. In fact, X1, X2, ..., X_r may be replaced by any r independent linear functions of themselves, and the c's will then be transformed by a linear substitution containing r^2 independent parameters. This, however, does not alter the type of group considered. For a single parameter there is, of course, only one type of group, which has been called cyclical. For a group of order two there is a single relation (X1X2) = [alpha]X1 + [beta]X2. If [alpha] and [beta] are not both zero, let [alpha] be finite. The relation may then be written ([alpha]X1 + [beta]X2, [alpha]^(-1)X2) = [alpha]X1 + [beta]X2. Hence if [alpha]X1 + [beta]X2 = X'1, and [alpha]^(-1)X2 = X'2, then (X'1X'2) = X'1. There are, therefore, just two types of group of order two, the one given by the relation last written, and the other by (X1X2) = 0. Lie has determined all distinct types of continuous groups of orders three or four; and all types of non-integrable groups (a term which will be explained immediately) of orders five and six (cf. Lie-Engel, iii. 713-744). Self-conjugate subgroups. Integrable groups. A problem of fundamental importance in connexion with any given continuous group is the determination of the self-conjugate subgroups which it contains. If X is an infinitesimal operation of a group, and Y any other, the general form of the infinitesimal operations which are conjugate to X is t^2 X + t(XY) + --- ((XY)Y) + .... 1.2 Any subgroup which contains all the operations conjugate to X must therefore contain all infinitesimal operations (XY), ((XY)Y), ..., where for Y each infinitesimal operation of the group is taken in turn. Hence if X'1, X'2, ..., X'_s are s linearly independent operations of the group which generate a self-conjugate subgroup of order s, then for _every_ infinitesimal operation Y of the group relations of the form __e=1 (X'_iY) = \ a_(ie)X'_e, (i = 1, 2, ..., s) /__e=s must be satisfied. Conversely, if such a set of relations is satisfied, X'1, X'2, ..., X'_s generate a subgroup of order s, which contains every operation conjugate to each of the infinitesimal generating operations, and is therefore a self-conjugate subgroup. A specially important self-conjugate subgroup is that generated by the combinants of the r infinitesimal generating operations. That these generate a self-conjugate subgroup follows from the relations (iii.). In fact, ((X_iX_j)X_k) = [Sigma] c_(ijs)(X_sX_k). s