508 Prof. K. Pearson. Generalised Theory [Sept. 11, 



on the generalised pure gamete theory ; and by 



on the theory of linear regression. 



In other words, regression holds for the great Imlk of a Mendeliun 

 population, but grows gradually less as we treat the nmtings of nearly 

 pure allogenic parents, ceasing entirely with absolutely pure allogenic 

 parents. When n = 5, a mating between pure allogenic parents 

 occurs only once in a million matings. If we limit our attention to 

 99 per cent, of the population, even when n is comparatively small, 

 we find regression on the mid-parent sensibly given by the Galtonian 

 theory of the midparent, and when n is anything at all considerable, 

 the Galtonian theory gives the value of the average offspring sensibly 

 correct for 999 in 1000 of the population, becoming absolutely correct 

 as n is indefinitely increased. The theory of regression shows us that 

 the planes (ii) are the best fitting planes to the hyperloloids (i), and 

 this fit becomes better and better as n becomes larger. 



Except for very small values of n, it would probably be impossible 

 to get an accurate test of the truth of the theory of the pure gamete 

 by using formula (i) ; the errors of random sampling are for the great 

 bulk of the population as large as the divergence between (i) and (ii). 



(j). If a correlation table for parents and offspring be formed in a 

 Mendelian population, while the regression will be real arid linear, the 

 variability of the number of allogenic couplets in the array of offspring 

 due to a parent of definite allogenic constitution will steadily increase 

 if we pass from the pure protogenic to the pure allogenic parent. The 

 mean variability of the arrays of offspring is equal to a- J (1 - r 2 ), 

 where <r is the variability of the general population, and r the 

 coefficient of parental correlation. This mean value exactly agrees 

 with that given by statistical theory. But no one has hitherto 

 observed this gradual change of variability in the arrays as we pass 

 across a parental correlation table. It is not very marked, especially 

 when n is large, and may, perhaps, have escaped notice ; still one would 

 be rather surprised if it had. This change of variability of arrays 

 of offspring seems to provide a method of finding n the number of 

 couplets in the constitution of the zygote. This point will doubtless 

 receive attention, and there is ample material already collected to test 

 it upon. 



(k). The frequency of, an array of offspring due to a given parent 

 depends upon the product of two skew binomials. Whether there is 

 approach to any skew, or normal curve, when n is increased has not 

 yet been investigated, but the deviation from normality exhibited 

 would on the surface appear to be considerable, and such deviation 

 would be inconsistent with the approximately elliptic contour lines 

 which have been noted by Gal ton in discussing human characters. 



