JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. ONE of the chief problems of mathematical statistics is

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... 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 observations, 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 very 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 expressiing " 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 surn, 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 incom-plete moments of the normal curve are important particularly in regard to plural partial 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 wvhich 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 monments, and the present paper is an attemnpt to deal with the problem.