10 On the Probable Errors of Frequency Constants 
We require accordingly a- a <rb r ab- Multiply the above expressions for 8a and Bb 
together, summing and dividing by the number of samples we find 
<r a o-br a b = ~ -fi'- (1 - r\ y ) (xxxvi). 
-tTTi ii o-„vl-r 2 L (x — x) 2 }* , ... 
Whence we deduce a T . =— — == — + - s — > (xxxvii). 
We note the increase of inaccuracy of the means of arrays far from the mean of the 
whole population*. 
The above equations embrace the chief results for the probable errors and 
errors' correlations of systems of two variables. They have been reached inde- 
pendently of any system of Gaussian distribution. 
* Eesults (xxxv), (xxxvi) and (xxxvii) are published here for the first time. 
