290 On Generalised Tchebycheff Theorems 
I have considered very carefully the possibilities of deducing higher ^'s from 
lower g's for non-normal systems on various hypotheses as to the nature of the 
regression and the scedasticity. The simplest hypothesis is to suppose linearity of 
regression, homoscedasticity and homocliticity of both sets of arrays. 
Let = 'Sf and = 
as usual ; let a single dash mark the /9's for the y variate, and double dashes the 
/3's for the y arrays of a;'s and triple dashes the yS's for the x arrays of y&. Then 
if y^^ be the mean of the .x-array of ?/'s, 
where y' is measured from the mean of the array, S is the sum for all members of 
the array and 2 the sum for all arrays. Thus if y^^ x be the regression line, 
since S , S — , is to be the same for every array. Thus 
'Hx tlx 
^0r-^^^' (ix). 
Similarly ^'("^l^Jl'l^J^. (x). 
(1 - r^f 
Thus it is impossible in homoclitic systems for the skewness of the arrays to be 
equal to the skewness of the marginal totals if there be correlation *. 
Again we have 
or 
+ %r"- (1 - r=) + /3/" (1 - rj, 
{l-r-'-r 
or, agam, /3o - 3 = (i ^ 
and snnilarly, p., — d = _ ^^y. \^^^h 
* We note that if the marginal totals be both without skewness, all the arrays will also be symme- 
trical. Equations (xi) and (xii) show us that if the marginal totals be mesokurtic the arrays will also 
be mesokurtic. 
