The first chapter contains an account of the extension of
Dirac's equation to general relativity while the second one gives
a summary of the generalised two - component spinor theory and its
application to the wave equation. Spinors are used extensively
in Chapter III to deal with the invariant theory of Dirac's
equation. Here certain results of Prof. E.T. Whittaker are
directly extended to general relativity and the complete scheme
of the simpler tensorial quantities including all those with
physical interpretations is developed, all the expressions and
the relations they obey being derived in a perfectly general
manner. A number of these relations are already known but now
all of them are proved without the necessity of referring to a
special coordinate system or of utilising a special set of matrices.
The vector form of the wave equation valid in all
space-times is derived from the spinor theory, agreeing in form
with the vector obtained by Prof. Whittaker from the special
relativity equation. In this formulation the wave equation
is expressible in terms of four null world-vectors which can replace
the 'k -functions, and all the tensorig quantities are
restated in terms of these vectors alone. The tensors and vector
wave equation are written out in detail in the case of a
Galilean system and these are expressed in matrix notation by
means of a special set of α-matrices. It is shown that the
matrix with imaginary elements is distinguished from the ones
with real elements in this form of the wave equation and the
effect of similarity transformations is considered.In Chapter IV it is shown that the criticism directed by
T. Levi -Civita against the Dirac system in that it depended for
its generalisation on specially distinguished directions in
space time, does not hold. In the first place his considera
:tions were really applied to an equation where the )v-function
was a world vector and so was not the usual wave equation and
secondly, the argument does not hold when one deals with the
actual Dirac equation which, because of the possibility of spin
transformations is shown to distinguish no special directions.The eigen functions for the hydrogen electron in momentum
space are found in Chapter IV, these are a finite series of
hypergeometric functions which do to elementary
functions. A form of the wave equation in momentum space is
used to derive the fine structure formula
