1t= n r=-xtyt a-, y n t-where there are n pairs of variables, such as xt, Yt, measured from their averages, is generally given as This was only proved to be correct when the joint frequency surface of x, y is normal in Professor Karl Pearson's original study; but Professor Edgeworth showed that an artificial normality could be produced by breaking up verynumerous observations into batches,'so that the same standard deviation was approximately obtained if the x, y surface was moderately abnormal. The full form of this standard deviation as given by Dr. Sheppard is r2 {M22 M40 M04 M31 M13 M22 n r2a2. u2y 4a4 4a4y ra3zay ra, a3y 2a2o2v where M8= mean of 8y,= IxytI in the notation used in the sequel, for all values of s and t.