With the help of a natural generalisation of the invariant scalar product for two spinor functions the invariant Fourier transformation of a spinor function can be defined, apart from a normalising factor. Assuming this factor as unity, the Fourier transformation of the solutions of Dirac's wave equation and its reciprocal are derived. The construction of reciprocal spinor functions leads to a transcendental equation for µ = ab/ħ which differs from that of the scalar case; but its roots are very similar to the latter.

Professor E. T. Whittaker has recently discovered a Third Quantum-Mechanical Principal Function R(q, Q, t - T) and has worked out the theory of this function in detail when the Hamiltonian is

By using the Sturm-Liouville theory of linear differential equations and the properties of Green's function, it is shown that the function is an elementary solution of the adjoint of the Schrodinger wave equation associated with the Hamiltonian H.

It is pointed out that the modified Planck constant ħ arises solely from the commutation relation and may, from the analytical view-point, be any constant, real or complex. In particular, if ħ = i, the use of an algebra with the commutation relation leads to an elementary solution of the real equation of parabolic type

It is shown that the elementary solution Γ(x, ξ; t – τ) of the equation

behaves, as t → τ + O, in very much the same manner as the elementary solution of the equation of heat.

In this paper the equations of the meson are treated in the same manner as the Dirac equation in a previous paper (K. Fuchs, 1940, Part IV). We use the formulae developed by Kemmer (1939), with a small modification, introduced so that the set of Maxwell's equations for the electro-magnetic field is obtained as a special case.