THE INCOMPLETE MOMENTS OF A NORMAL SOLID. 
By a. RITCHIE-SCOTT, D.Sc. 
1. Introductory. 
One of the chief problems of mathematical statistics is the representation as 
closely as possible of a series of observations by a mathematical formula, or speaking 
graphically, by fitting a geometrically determinate curve or surface to the obser- 
vations, these being represented by some kind of space unit. The formula, curve or 
surface constitutes a resume of the known facts and gives that conception of them 
which is the scientific law. underlying their relations. The veiy conception, however, 
of the statistical method involves a classification of the phenomena into groups, which 
assumes identity for the purposes of classification and ignores the infinitely fine 
gradation within the group. We have therefore to deal, not with the actual 
measurements, but with groups of measurements labelled with some representative 
value or class mark and to these groups we must fit our formula, curve or surface. 
In the process of fitting, the method of most universal application is the method 
of moments which consists of expressing "the area and moments of the curve or 
surface for the given range of observations in terms [of the real constants of the 
theoretical curve] and equating these to the like quantities for the observations " 
(K. Pearson, "On the systematic fitting of curves," Biometrika, Vol. i, 1902, p. 270). 
In the early history of mathematical statistics Gauss fitted the normal curve to 
observations by means of zero, first and second moments, i.e. by means of the sum, 
mean and standard deviation of the observations. The method of least squares is for 
any method of polynomial fitting a method of moments in which high moments 
may have to be used. 
It was soon discovered, however, that the incomplete moments of the normal 
curve are important particularly in regard to plural jjartial correlation and the fitting 
of incomplete curves, while the development of the ideas of multiple correlation and 
variation brought into view the need of multiple moments and multiple product 
moments which are still further required in the evaluation of the probable error of 
multiple correlation coefficients. 
With multiple variates we have the same problem as with the single variate, 
viz. the reconstruction of a population from a portion of it, and for this purpose 
incomplete moments and product moments are essential. Further, the theory of 
plural partial multiple correlation of observations classed in broad categories depends 
entirely on a knowledge of these incomplete product moments. 
It is therefore from several points of view very desirable to obtain algebraical 
expression for these incomplete moments, and the present paper is an attempt to 
deal with the problem. 
Biometrika xiii 2.5* 
