8 On the Probable Errors of Frequency Constants 
(7) To find the correlation between a mean and a coefficient of correlation, 
i.e. between x and r X!t . 
We have to multiply $x — 8p lti with 
8r x y _ Bpn _ 1 8pm _ 1 Spm 
r X!l pn 2 p 20 2 p 0 , ' 
and apply (xxv) to each term. We have 
>••• x ''■ r -"" t N[pn • 2^20 2jp 02 
_ 1 (r X yO) l a^/p 1 1 V/^ov* 1 r Xil <j x a y -\l '&') 
-Zv I r xy <Ty<T x 2 ov 2 er ly 
iV 
using the values of j% and j? 12 for linear regression. If we now use the value 
in (xxi) for a., we have 
1,. [ 
7 = - — — ; • r (xxix) 
V. 
(l-r%) l-i(/3 2 -3 + /3./-3) 
( x - 
reducing- when the kurtosis of both distributions is zero to 
lr w |Vft-r,Vtt) 
'«* 2 1 - /•%, 
and vanishing for all symmetrical linearly correlated variates, including of course 
Gaussian systems. 
(8) To find the correlation between a deviation in a standard deviation and 
one in a coefficient of correlation. We have to multiply 8p w by 
8r xy _ 8pn _ 1 8p^ _ 1 Spv 
r xy pu 2 p, t) 2 j9 02 ' 
We find 
1 PSI-P^PU 1 P40-P20 2 Ipm-PioPo 
— a„ a,, r, 
ry IH r xy /J-., >' x y N p n 2 JS% 0 2 Npv ' 
Hence, assuming linearity of regression, we may put 
psi/pu = <rx/3 s , pjpoo = (3 2 <r x 2 , 
and approximately by the result immediately above Equation (xvii) 
P22/P02 - p*> = <r x *r\ y V(/3 2 - 1 ) 08,' - 1 ). 
Thus 
1 o-/V/3 2 -l 1 
7f - = w {*£ (& - 1) - fa 08, - 1) - to^y V(& - !)(&'- 1)}. 
Or cr_ r ff = ^= [V& - 1 - rU V/8,' - 1}, 
