200 
DR. W. F. SHEPPARD ON 
Appendices. 
Appendix I.—The correlation-determinant. 
,, II.—Frequency of-correlated errors. 
„ III.—Improved advancing differences in terms of sums 
,, IV.—Formulae in terms of u’s . 
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I. Introductory. — This paper is a development of two earlier papers,* which for 
brevity I call “ Reduction ” and “Fitting” respectively. The paper f immediately 
preceding “ Fitting ” is referred to as “ Factorial Moments.” 
These earlier papers deal with two problems, which are closely connected and have 
the same solution. For both of them, the data are a set of quantities u 0 , u u u 2 , ... of 
the same kind, which we regard as representing certain true values U 0 , U 1} U 2 , ..., 
with errors e 0 , e u e 2 , ..., so that u r = U r + e r . These errors may be independent or 
may be correlated in any way. The first problem is based on the assumption (which 
defines the class of cases we are dealing with) that the sequence of U’s is fairly 
regular, so that differences after those of a certain order, which we will call./, are 
negligible. This being so, we may alter any u, or any linear compound of the us, 
such as an interpolation-formula, by adding to it any linear compound of the neg¬ 
ligible differences. (I use tha term “linear compound” in preference to “linear 
function,” since there is no consideration of functionality.) The problem is to find 
the value of the resulting sum when, by suitable choice of the coefficients in the 
added portion, the mean square of error of the sum is a minimum. This is the 
problem of “reduction of error.” For the second problem it is assumed that U r is a 
polynomial in r of degree j, and the problem is to find the coefficients in this poly¬ 
nomial by the method of least squares. This is the problem of “ fitting.” 
The practical solution of these problems for the general case, in which the errors 
are correlated, is not easy. The particular case which is simple is that in which 
the errors all have the same mean square, which by a suitable choice of unit is taken 
to be 1, and the mean products of error are all 0. (In the previous papers I have 
called this system of errors the standard system ; in the present paper the set of us 
which possesses this property is called a self-conjugate set.) In “ Reduction ” I gave 
the solution for this particular case in terms of central differences, and in “ Fitting ” 
I gave the solution in terms of advancing differences and of advancing and central 
sums, formed in a particular way. I also gave expressions in terms of the us, but 
these were rather complicated. It remained to obtain expressions for the mean 
squares of error of the new values, in order to compare them with those of the old 
* “Reduction of Errors by means of Negligible Differences,” ‘Fifth International Congress of 
Mathematicians,’ Cambridge, 1912, ii., 348-384; “Fitting of Polynomial by Method of Least Squares,” 
‘ Proceedings of the London Mathematical Society,’ 2nd series, xiii., 97-108. 
t “ Factorial Moments in terms of Sums or Differences,” 1 Proceedings of the London Mathematical 
Society,’ 2nd series, xiii., 81-96, 
