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The short note by Duffin contains much of the matter here more fully developed. The writer was studying the same subject previously, but is glad to acknowledge that Duffin's statement of the commutation rules was new to him and has greatly influenced the detailed development of the work. Many thanks for exchange of information are due both to Dr Duffin and to Dr Belinfante.

In an interesting recent paper Schrödinger(5) contributes much new information on the properties of the hypercomplex algebra used in meson theory(1, 2, 4, 6, 7) which is defined by the relations(3)

1. Introduction. The theory of heavy electrons recently developed by several authors')" may be considered to give a satisfactory account of the empirically known neutron-proton interaction. However, it now seems well established that there exists a proton-proton interaction of comparable magnitude which is not accounted for equally well. Owing to the fact that the emission of a heavy electron involves the change of a neutron into a proton or vice versa, the first approximation of this theory gives only an exchange force between unlike particles, whereas a force between like particles must be due to double transitions and thus only appears in the second approximation. It is true that the expansion in terms of the number of particles emitted is actually so badly convergent that the second order proton-proton force at distances of about 10~1 3 cm. is found to be not essentially smaller than the first order neutron-proton force (see FHK), but nevertheless this does not explain experimental facts, since the calculated second order force is always repulsive.According to the measurements of Tuve, Heydenburg and Hafstadtf the proton-proton force is definitely attractive, and also apparently is very nearly equal to the neutron-proton force in corresponding anti-symmetrical states. The latter fact has served as the basis of the hypothesis that the forces between the elementary nuclear particles are essentially independent of the charges of the particular particles concerned. In the following work we denote this "chargeindependence hypothesis" by the abbreviation CI-hypothesis. According to this view the observed proton-proton interaction should be exactly equal to the corresponding neutron-proton force, and the neutron-neutron force, which cannot be so directly measured, should also be exactly the same. In the theory of binding energies this hypothesis has proved very useful, particularly because it has made it possible to determine the exchange character of nuclear forces practically uniquely.

Explicit ~atrix. eleme~~s are foun~ !~~ the ,generators of the ~roup R(5) in an arbitrary irreducible representatIOn usmg the natural basIs m whIch th.e represen.tatIon of R(5) is fully reduced with respect to the.subg~oup R(4) = .SU(~) @ SU(2). ,!,he t~chnIque used IS based on the well-known Racah algebra. The dlmen.slOn formula ~s derIved and the mvarIants are found. A family of identities is established which relates varIOUS polynomIals of degree four in the generators and which holds in any representation of the group.