338 On Lmearity of Regression 
(1 - r"f . 
Now the formula for S,.- consists of a term — involving r only together 
with a number of terras involving product moments which are just such terras as 
the product moment terms of equation (xxii)*. Again the formula for consists 
of a term ^ ^ involving rj only together with a number of terms of exactly 
the same nature as these moment terms t- In practical statistical work the simple 
(1_ 7-2)2 H— 7l2)2 
formulae %r = — ^ — , = - — ^ are used and the nature of the justification 
for these simple formulae can only be appreciated by a reference to the numerical 
work of Karl Pearson's memoir where he shows that, as a matter of experience, 
in actual frequency distributions which are very far from normal, although the 
separate product terms occurring in the formula for S,,^ may be significant, yet, 
these product terms occur in such a manner in the formula as to be insignificant 
in the aggregate. This work suggests that we should consider if anything of a 
like nature occurs in the case of the formula for 2^-. Equation (xxii) gives as 
the sum of a term involving rj and r only and a number of product 
moment terms of exactly' the same type as those occurring in the formulae for 
2r", 2^-. I suggest therefore the formula 
' N{r 
.(xxiii), 
as being exactly analogous to the simple formulae for 2,.S 2,-; the complete justifi- 
cation for the formula (xxiii) will be found in the practical statistical work given 
later in the paper. 
Before proceeding to the numerical work I will give some formulae which 
enable us to determine other quantities of importance when 2^-" is known. 
Problem V. To find the correlation between the deviations due to random 
sampling in rj and r. 
We have 2^= = % + ^ - 
7)- r- rjr 
V V \2 9V •< 
7] r j 7]r 
22 2 /'^ \- 
or ^'-(1 - K) = - - ^) (xxiv). 
This equation gives the numerical value of i?,,. as soon as r, tj, 2;., 2,,, 2^ have 
been determined, no matter in what way. 
Assuming the simple formulae for 2,., 2^, and the formula (xxiii) for 2^, we get 
a corresponding formula 
2r}r (1 - ,;-^) (1 - r^) (1 - R^,) = rj'^ - r"- - (77 - rf (1 + r^rf (xxv). 
* Phil. Trans. A, Vol. 191, p. 245. \ Loc. cit. p. 20. 
