Miscellanea 
251 
Let the new system of variables be : 
Now on the hypothesis of both frequencies being random samples of the same population : 
fp 
fp 
N 
iV 
M 
fi 
N 
N' 
M 
and the mean values, x p and x v will be zero. 
Further let <r p and <j q be the standard deviations of x p and x q , and r vq the correlation of the 
latter pair. Then bearing in mind what has been said about the independence of the frequencies 
in the two series, we have at once : 
%+lfc) 
(v). 
yjr* T N't 
Whence we find from (i) and (ii) 
^K* + f)*B/) (vi ^ 
WV,«-« 2 (^ + ^)%T ( vii )- 
Now write /V= '> 2 ( L + 4^ ) 1? ( viii )> 
^ W (i+^)S (ix) > 
and M ' = n \N + W') (x) " 
Then ix p 'jM' = nj M, 
and since (fi p ) = M, it follows that Si s (n P ') = M', or J/' is the total population of a frequency 
distribution 
Ml', H2, M:l'> ••■ h>'> /V> •■• P»- 
Further o> 2 =/*»>' H -jjbj (»), 
<r P <r q r m = - -jjfi- (xii). 
Now (xi) and (xii) are absolutely identical with the type-standard-deviation and type- 
correlation of the frequency distribution of the system a? 3 , ... x p , x q , ... a?, as measured 
from a theoretical system /*/, ^ 2 '> ••• ft,', M«\ ••• /*»■ They agree in form with equations (vii) 
and (viii) of my paper " On the Criterion that a given System of Deviations from the Probable 
in the case of a Correlated System of Variables is such that it can be reasonably supposed 
to have arisen from Random Sampling," Phil. Mag., Vol. 50, 1900, p. 161. 
Hence we have only to form 
