6 On the Probable Errors of Frequency Constants 
This result is of much interest. If the kurtosis be zero, then 
/3 2 - 3 = 0 = &' - 3, 
1 — r" 1 
and we have a,. = — (xxii), 
the value originally given by Pearson and Filon for the standard deviation of 
a correlation coefficient when the frequency surface is Gaussian {Phil. Trans. 
Vol. 191, A, p. 242). We see accordingly that: 
(i) Equal kurtosis is needful in the two variates if the regression is to be 
linear and the arrays to be homoscedastic in the case of each variable. 
(ii) The ordinary value subject to (i) is only correct provided the kurtosis 
is zero, and this is true whether the distribution be Gaussian or not. 
(iii) The ordinary formula may give very inaccurate results, if the kurtosis be 
considerable and the correlation high. 
(iv) It is probable that (xxi), as we have taken a mean value for p i2 , gives 
fairly good results even when the correlation is not linear. 
Clearly we must always have 
2 . .... 
r xv < - j — (xxm). 
Or, for linear regression in homoscedastic systems there is a superior limit to the 
correlation possible with given values of the kurtosis. This is an interesting point, 
and forms a remarkable limitation on the nature of double linear homoscedastic 
regression. 
(4) We may now find the correlation between a product-moment <p q ^ about 
fixed axes and p u ,u' a product-moment about axes through the centroid. We have 
to multiply together (vii) bis and (xiii). We have 
_ S mfimf x (x s - (y, - y)4 
- up u ^, tfte (ar, - x) ccSyA 
- upu,u^ S & (y, - y) tf/y/} (xxiv). 
We can simplify the form of (xxiv) by taking the fixed axes now through the 
centroid itself. This gives us 
a Plhtl . a P ' r p q , v >p u , u > = ]y -Pg,*Pu,« ~ »T>u- h vp q+hq ' - rfpu,vf-ipq,t+i} 
(xxv). 
