One of the many ways of obtaining from a set of
observations a second smoothed or graduated set is to
assume that the second set is a linear combination of the
first. Thus if u denotes the column vector of n observed
values, y that of the graduated and C the matrix performing
the linear transformation, then y = Cu. This method was
considered by 7.F. Sheppard, in the case where the
observed data are equidistant, equally weighted and
uncorrelated; the assumptions being that the sum of the
squared coefficients in the transformation shall be a
minimum; and that each / shall differ from a specified u
by differences of u of order exceeding j, i.e. if the u's
are already polynomial values of degree j, then the linear
transformation leaves them unaltered. In this way each
graduated value depends upon every observation, and not
simply on those on either side as, for example, in the
case of the centred finite summation formulae of Spencer
or Woolhouse. Sheppard points out that the solution
of this problem yields precisely the same final results
as that of fitting a curve of degree j to the u's by the
method of least squares. A.C. Aitken has shown more
recently how this problem in its two aspects may be solved
much more concisely by using the matrix calculus, and
indeed he gives the solution for the case where the u's
are not subject to the above restricted conditions but
may be of arbitrary functional type. The transformations
which he derives for the restricted and general cases are

y = P (P' V' P)⁻'p' (no correlation and equal weights)

and y = P(P' V' P)⁻'P' V⁻ respectively,

where P is a matrix of prescribed functional values by
which the y's are expressed, and V is the symmetric
variance matrix associated with the data u. (V[vij]:[Pijδiδj])

In Chapter I of this thesis the problem of graduation
by linear combination is again considered, but with
different minimal conditions. Firstly, what linear
combination y - Cu is such that the set of k differences
Pky = CIA, has minimum sum of squared residuals and
secondly, what linear combination CDcu of the k differences
-05 of the observed values produces a set of smoothed k
differences with minimum sum squared residuals. Examples
are given using both factorial polynomials and the orthogonal polynomials of Tchebychef. It is also shown that
this problem leads to the same solution as that obtained by
using Sheppard's original assumptions.

In Chapter II the linear combination C of observed
data is considered where the y's are expressed in terms of
the harmonic functions. The properties of the transforming
matrix are established, and the Fourier coefficients
are given in matrix form. The question of estimate
errors from residuals which is of prime importance in the
examination of any physical phenomena associated with
harmonic analysis, is also considered.

In the appendix tables of C are given for values
of 2n, the number of data, equal to .,6, 6,10,12,16 and 24
with k = 1,2 . n, the number of harmonics in the series.
A bibliography of works consulted is also given.