68 PROFESSOR K. PEARSON ON A GENERALISED THEORY OP ALTERNATIVE 



centimetres, &c., and each such unit will not, as a rule, represent one allogenic 

 couplet ; but interpolation ought to give a result not widely divergent from the 



truth. 



The method would of course fail practically if n were very large. For example, if 

 n were 48, the deviation of the required group of fathers would be 3(r, and hence 

 such a father would only occur once in 1000 individuals. In a manageable population, 

 therefore, we are very unlikely to have enough such fathers to form any reliable 

 measure of the variability of their offspring. At the same time, the squares of the 

 variabilities of the arrays of sons due to quite frequent fathers ought to give a 

 straight line, and if this line be determined properly, there should be no difficulty in 

 finding the theoretical position of the above father, and so finding n. 



Many other physiological theories besides the present might give this peculiarity 

 of the diminishing variability of the arrays of offspring as we pass from one side of 

 the correlation table to the other. Such changes in variability are familiar to those 

 who have had to deal with skew correlation. But, as far as we are aware, they have 

 Hot hitherto been noticed in inheritance tables. The existence of this changing 

 variability would not affect in any way the general theory of linear regression applied 

 to heredity in populations. It would, however, lead to an immediate extension of 

 that theory consisting in the tabling of the standard deviations of the arrays. 

 Should the standard deviations of these arrays show no bias towards a linear 

 distribution, but only the fluctuation to be expected from random sampling about 

 the mean value cr */ (l r-), we should have a strong argument against the present 

 general theory of alternative inheritance. We seem here, therefore, to have a crucial 

 test of the validity of the theory, which may be quite as easy to apply as the, previous 

 test of the numerical value of the parental correlation. 



Of course the results now reached are not consistent theoretically with normal 

 correlation surfaces with their elliptic contour lines. The fact that Mr. GALTON came 

 to his elliptic contours in the first instance on the basis of his observations, and not 

 from any theory,* shows that they must in the case he was dealing with be approxi- 

 mately correct. Further, there is no doubt that in other statistics for characters in 

 man there is within the limits of random sampling a close approximation to normal 

 distribution. It might be hard to consider that such a deviation as would arise with 

 a continuously increasing variability of the arrays from one side to the other of the 

 table could exist and escape notice, had we not in physics had evidence that theory 

 has often led to the discovery of an obvious relation which time after time must have 

 been overlooked by previous observers unprovided with the theoretical hint of what 

 to seek for. Hence while we may say that the parental correlation given by the 

 theory is too rigid for the facts, we must leave this second test until more careful 

 examination ad hoc has been made of the ample existing data. 



* 'Natural Inheritance,' p. 101. 



