Karl Pearson 
291 
.(xiii) 
Now consider g'22, 
= ^,r^ +(1 
by symmetry. Hence it follows that in linear homoscedastic systems = /3„', and 
accordingly 
- 3 = - 3 = - 3) (xiv). 
This is of interest as indicating that in linear homoscedastic systems the one with 
mesokurtic margins is the only one in which the kurtosis of the arrays can be the 
same as that of the margins. 
Agaia q,, ^^S(a^f)/{cT^a,') = ^ tSx' x + y'J^ «7,V./ 
SrX{n,x^) / 3/'M 1 /n,x>\ ( \ 
= r8/3, + 3r (1 - /3, + s/ fi, V/S,' - r^A i^^) 
= r^/S; + 3r (1 - r'O + V^? V^"^ - ^''/S/ (xvi) 
by symmetry. 
It follows from (xv) and (xvi) that it is needful for 
A-/3, = /3;-/3i' (xvii). 
Finally we l^ave 
q^ = ^S(af^y*)/(a,^a,^) 
= ^X8x^ (r^'x + y) /cT.^cr, 
= r^y3e + 6r' (1 - r') /3, - ^r"-^, + 4>0, ^13//^, + A/3/" (1 - r% 
or 
qu = ''^/Q^ + 6r"-{l - j-^) /3, - + 4/3, V/S/p; - r^/3,= - 6r= (1 - r'^) ^ + AA' 
= r^/S; + 6r' (1 - r^) yS/ - ^r'/3; + 4/3/ v'/3,7^' - r'^,'' - 6r'(l - r') /3; + ySo'/S, 
(xviii), 
which again involves the complicated /3-relation 
(A - yg/) + (1 - 1-) (A - A') - 47- (/S^ - A') + 4 {/33/3,' - A'^,) V/3,/3/ = 0 
(xix). 
It is difficult to see how the form of variation of one character can be related by 
the correlation between that and another character to the form of variation of the 
second character as (xix) would indicate. If it were we should get into great 
difficulties in dealing with similar conditions to (xix) for a large number of characters 
