The group (F) which is the object of the present investigation has already been briefly discussed by HuRWIEZ.t His paper deals chiefly with the groups of signs (0, 3; 2, 4, oc) and (0, 3; 2, 6, cc)4 He proves them commensurable with the modular-group (0, 3; 2, 3, oc) and hence derives certain algebraic relations between the simplest functions belonging to his groups on the one hand and the invariant J of the modular group on the other. He also obtains the arithmetic character of the substitutions of his groups. The group r is also mentioned briefly in FRICKE-KLEIN'S treatise, ? where it appears as the reproducing group of a certain ternary quadratic form. In Part I, we derive r as the monodromy group of the Riemann surface R (p = 2) defined by the equation