Karl Pearson 
411 
To do this we require to know the value of 
Now it clearly equals 
Vx 2 2 a? 3 2 x s ¥x" 
TT + "W + ~W + TP + 
x ~ (x ^ e x - ; ■ h i 
Let m — —, then 
m 
n 
l 2 n 2 2 fn\ 2 3 2 / nV 4 2 , n \ 4 
me rn. 1— — 4 — _ 4 _ / _ 4 1 _ + 
[1! m 2! \w 3! \m/ 4! mJ 
— S (s"pg) by the first equation of (xii) 
n . . n 
— n n , \ — 
= ?ne m — hie™ 
= n 4 = N, to our degree of approximation. 
For example 11 (1 4 — ]=3627'6 in the enteric data cited above, = 3627 - 4 
actually, when calculated as S(s 2 p x ), a difference of no importance. 
Now let us write x s = s"-p s ; then clearly cr 2 a . = s-a 2 p , and R x = R p , for the 
new variables only differ from the old by constant multipliers. Accordingly 
we have 
' \ 
2 - (1 x - 
*• s V N 
a x s a x t n x s x t - - -Jf 
.(xix). 
while S (x s ) = N 
Thus «i, cc 2 , ...x s ... may be looked upon as frequencies, which have a constant 
sum N and for which the standard deviations and correlations are precisely those 
defined in my original memoir on " Goodness of Fit " : see Phil. Mag. Vol. L. 
Eqns (vii) and (viii), p. 161, July, 1900. We are not troubled by p 0 for it does 
not enter into the system (i.e. x 0 = 0), and p 0 is merely an additional quantity 
to be found ultimately by the relation 
n 
p 0 = m — S 
Proceeding exactly as in the memoir just referred to, we find 
^ i *<P.-m (xx> 
A 1 x s r { Ps. J 
It will be observed that this result differs from the x 2 f° r a series of frequencies 
p 0 , j>i, ■•■Ps---, i-e. from 
o (p* - p*y 
