840 
On Li7iearity of Regression 
Also we have 
+ 2*5 [nx^y^ {x, - {yx^ - y) (ys - yx)] 
hence p,n2 = X,^, + -Sf {nx^^n^"~ («p - (xxxi). 
Thus the formula for contains only X.'s and which, by their definition, 
are calculable when Hx^, yx^, known for every array, together with product 
moments of the form p,„i, which are also calculable if we use equations 
(xxx) and (xxxi). 
In his memoir Karl Pearson discusses four actual frequency distributions 
which exhibit skew correlation and which he names illustrations A, B, G, D. For 
these distributions he has given the numerical values of many of the statistical 
constants and has also tabled the 7ix^, yx^, o'n^ < ^'^^ each array, to four decimal 
places in the cases of illustrations A, B, C, and to three decimal places in the case 
of illustration D. It is thus relatively easy to calculate Sxb- from the formula (xxii) 
for these four cases. Starting with the tabled values of 7?^^, y^^, (Tn^ , I recalculated 
all the constants necessary, keeping six decimal places throughout the arithmetical 
processes and using Sheppard's corrections or not, according as Karl Pearson 
did or did not use them in his work. I have tabled all the constants to four 
decimal places, but, whenever any constant was used to obtain a new quantity, 
its value as obtained by me to six decimal places was used. 
Although Karl Pearson obtained his constants by a different arithmetical 
process, and his results are necessarily more accurate, since he has kept six decimal 
places thi'oughout, yet, wherever the values of constants given by my work may 
be compared with his, they agree to at least four decimal places excepting in the 
case of illustration C, in which the general agreement is only as far as the third 
decimal figure. This illustration has given throughout more irregular results than 
the others, due probably to the disturbing influence of the ecdyses. 
Having obtained the necessary constants I proceeded to calculate S^- for each 
case from the formula (xxii). 
It remained to test the closeness to which the simple formulae (xxvii), (xxix) 
might be used instead of the formulae (xxvi), (xxviii) for , 2:^. For this purpose 
Karl Pearson's four illustrations are again extremely convenient since these 
are the only published frequency distributions for which 2,, has been calculated 
from the complex formula. In each of the four cases I substituted in the formulae 
(xxvi) and (xxviii) for 1^, 2^ the values found from the complex formulae, and 
