18G Probable Errors in Multiple Skew Regression 
In the present paper the corresponding results are obtained for the case of skew 
regression. The method employed is different and by supposing the regression 
to be linear and the distribution to be normal, a confirmation is obtained of the 
above results which in Pearson and Filon's memoir depend on very complicated 
analysis. 
(2) We may begin by discussing the correlation that exists between deviations 
from their means in the case of two correlation coefficients r^y and r^^. 
We have r^y = p^J^/{p^■,py■;) (8), 
^zt = PztlViPz'-Pl^) (9), 
where Pxhfz'H^ employed to denote the mixed moment of orders I, m, n, k in 
the variables, taken about the means so that 
^xy Pxy ^ Px^ ^ Pv^ 
and 
^zt Pzt 2 p^, 2 pf. 
It is clear that we shall require the correlations between any one of 2'>xv> Px^> Pv" 
and any one of p^^, p^i and pfi. It will suffice to find the correlation between 
p^y and p,t. 
Now Np^y = SS {n,y {x-^{y-y)} (12), 
X y 
.'. Ndp^.y = S S {dn^y {x — x) {y — y)} 
+ SS{-n,y {y - y) dx} + SS{- n,y (x - x) dy] (13), 
X y X y 
or Ndp^y = SS{dn^yXY} (14), 
X y 
if we denote the total population by N, x — x hy X, y — y hj Y and remember 
that 
S {(x - x) n^,} = S {{y - y) n^y} = 0. 
X y 
Similarly, Ndp,, = S S {dn,t ZT} (15). 
c t 
The mean value of dn^ydn^t in many samples is the mean value of 
where in the fourfold summation the term 
is omitted. 
But clearly the right-hand member of (16) reduces to 
hence the mean value of 
N~dp,ydp,( is N {p,,j,, - f^yf^t) (17). 
