270 PROFESSOR K. PEARSON AND MR, L. N. G. FILON 



Lastly, 



M 1:i = ",~^ T) > 

 whence 



This completes the direct series of probable errors and error correlations. By aid 

 of the above correlations and standard deviations we can now find a further series. 



From (xlii.) we have for the standard deviation cr (about the mean), &' = {*., = 



/> + 1 

 , ' - 



v 

 or cr = Hence 



A 5 , A,, A 7 



o- "~ r + i 7 ' 



Square both sides of this, divide by n and sum, we have at once from the definition 

 of a coefficient of correlation 



('v c- *?- v- P v v 



-- \ i _ ^ _ 4. -v _ *siz*a . 

 *+l S ' J ' +1 



70' +1) 

 Hence, using (xlix.), (1.), and (li.), we find, after reductions, 



Multiply (liv.) by A/;, sum and divide by n, we have 



^ 



o- " ! p + I 



Whence, by (Hi.), 



RA.T = --- 



7 s 



or, reducing by (1.), (liii.), and (lv.), 



Next, if S<; be the skewness, we have from (xlv.) 



