This research has been carried out on Dr Aitken's
suggestion. The first chapter is largely a recapitulation
of known results which I have learnt from Dr Aitken, here
arranged for convenience of reference later in the thesis.
The second chapter is the application of these methods
to the deduction of a two-variate GamMa type Distribution.
Dr Aitken pointed out that the variances in a normally
correlated two-variate distribution would give the required
distribution,and chanter II is just the carrying out of that
suggestion. fie also directed me to the papers by Hardy and
by Wishart and Bartlett which give rise to chapter III.
Those two chapters form the centre or core of the thesis
from which the other research radiates in three main directions, of varying interest from the points of view of pure
mathematics and of statistical applications.
The first is chapter IV which is purely of mathematical
interest. It contains the most substantial single piece
of research in the thesis. It was perhaps fortunate that
on my first searching Watson's Theory of Bessel Functions
for theorems involving incomplete Gamma functions, I did not
find the paragraph on Hadamadd's paper. The result given
by Watson would have been sufficient for the purposes of
chapter III, but not being satisfied with my own deduction
of it, I sought to find it as a special case of a more
general theorem, and in doing so have been led to discover
a more general result, and incidental results which may be
of great interest in themselves.
Secondly, the generalisations in chapters V and VI
are of interest as giving forms for statistical distributions,
but they wilt also be of interest for pure mathematics,
giving, for example; a generalisation to any number of
variables of Mehler s theorem (1866) which has been discussed
by Hardy and Watson and others.
The third branch of the thesis is,concerned with the
actual method of fitting a distribution function such as that discussed in chapters II and III. This is discussed
in Chapter IX, and tables to make easy the fitting by the
method suggested there of a type III curve are given in
chapter X.
Other parts of the thesis are chapters VII and VIII,
of which. VII is concerned with an attempt, not so far
successful, to extend the theory to all Pearson's types,
instead of type III only. (Similar attempts must have been
made before s see Romanovsky, Biometrika, Vol XVI, parts I
and II, p. 106) And chapter VIII applies a method whit I
learnt from Dr Aitken in application to a normal distribution
to the distribution of chapter II, and deduces simple formal
which may be of great practical significance in dealing with
problems of selection in distributions in which the coefficient
of variation is important and is not small
