292 
On Generalised Tcliehycheff Theorems 
with different correlations. If as it appears to me (xix) would need to be satisfied 
independently of r, then we must have 
A-6/3,-/3;-G/s; \ 
8/3,-2/3, = 3/3;-2;8;[ (xx). 
The second of (xx) by aid of (xvii) leads us to 
whence /S, = /3,', and as /3., = /3/ it follows that /3, = ^,', fi, = l3,', ^o = /3>', that is to 
say the total frequencies of the two correlated characters must possess variation 
practically of the same type. 
Now I find this is very far from being the case iti distributions which differ 
widely from the normal correlation surface. Thus it follows that the hypothesis of 
homoscedasticity, linear regression and homocliticity fails for such cases. I therefore 
modified the linear regression and adopted skew regression, homoscedasticity and 
homocliticity. I again got relations between the yS's, but of a much higher degree 
of complexity. These were tested by Mr A. W. Young and myself on the skew 
correlation surfaces of barometric data, but were found to fail. Direct investigation 
afterwards showed me that while the regression differed to some extent from 
linearity, it was the homoscedasticity which was in the first place the erroneous 
assumption. The arrays were very far from having the same standard deviations. 
Until therefore some theoretical advance is made in the investigation of skew 
regression surfaces, especially for those which have linear or nearl}' linear regression 
combined with hcteroscedasticity, it is unlikely that we shall have any adequate 
method of determining high product moment-coefiicients from low ones. We are 
accordingly thrown back on direct determination of the high product moment- 
coefficients, if we wish to determine a Tchebycheff limit. The work of determining 
It would involve a whole roimd of 8tb order moment-coefficients and product 
moment-coefficients. It would then give us a limit of the order '95 for '99. Lower 
order I's would hardly give values of much importance, and it may be questioned 
whether a rough limit of the kind required could not be better obtained by inserting 
the desired contour on a "scatter diagram" and simply counting the dots which 
fall outside it, or indeed by taking. the best fitting normal surface to the actual 
distribution. The reader may question whether something better could not be 
achieved for skew correlation Tchebycheff limits by some contour other than the 
ellipse. This would imdoubtedly be the case, if we knew the forms of the skew- 
correlation contours, for then we shoidd undoubtedly choose this equi-probable locus 
for our boundary. But as we have only a knowledge of these empiiically — experience 
shows them to be frequently pear or lemniscate loop shaped — we get little help for 
om- present jiroblem. ' 
One other aspect of the matter may be briefly considered. We may find a limit 
to the probability that an event or individual will lie within a circle of radius R 
