314 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
group m x by any series of selections and breedings) and m 2 their offspring-mean, is 
(M 2 m 1 — Mpn^/^Mo — m 2 — M x + m x ) constant for all stages of selection ? If it be, 
it is the stable focus of regression, and 1 — - -vris the coefficient of regression. 
i-'J-i ~ lYLg 
(f.) Progression of the Focus of Regression with Natural Selection. 
(i.) General Re?narks on Regression and Fixedness of Character. —Our first 
hypothesis certainly favours the general views of those who support the doctrine of 
panmixia, although to be quite consistent they must : 
(i.) Place the focus of regression back at the zero size of an organ or the zero 
degree of intensity of a characteristic. 
(ii.) Assume much nearer approximation to unity in their coefficients of regression 
than any measurement as yet suggests, or 
(iii.) Demand a far higher mortality of periodic natural selection than has 
anywhere as yet been demonstrated. 
Professor Weismann has no difficulty, apparently, about (i.): “As soon as natural 
selection ceases to operate upon any character, structural or functional, it begins to 
disappear (“Essays on Heredity,” 1889, p. 90.) He talks of functionless organs 
losing in size with the suspension of natural selection “ until the last remnant finally 
disappears ” (ibid., p. 292), while “the disposition of the tail to become rudimentary, 
in cats and dogs, may be explained in the simplest way, by the process which I have 
formerly called panmixia,” i.e., suspension of natural selection (ibid., p. 430). This 
explanation “in the simplest way” fails entirely to say whether (ii.) or (iii.) is to be 
accepted after assuming the truth of (i.). What is quite clear is that in the only 
case where either the coefficients of regression or the mortality can at present be even 
approximately stated neither (ii.) nor (iii.) hold. Fox-terriers and domestic ducks 
may be bred with a comparatively small mortality, but how great must be the 
coefficients of regression if their foci of regression are to be placed only as far back, say, 
as at general populations of jackals and wild ducks.* Apart from cases of atavism, 
which may be looked upon as improbable variations amply allowed for by theory, w T e do 
note, even in dogs, a regression towards a distant ancestry (Darwin: “Animals and 
Plants under Domestication,” vol. f, pp. 37, et seq.). In these cases, however, change 
of environment seems in some way more important than the suspension of natural 
selection. We have, so far, evidence in favour of Mr. Galton’s view of positions of 
stability for the focus of regression. It seems, indeed, to be a general opinion among 
breeders that a character can be fixed, a stock made to breed truer by repeated 
selection. 
Thus Darwin writes on “ Fixedness of Character :” “It is a general belief amongst 
breeders, that the longer any character has been transmitted by a breed, the more 
fully it will continue to be transmitted. I do not wish to dispute the truth of 
* Professor Weismann would place the focus of regression for domestic ducks much further back, 
presumably in a wingless stage. (“ Essays on Heredity,” p. 90.) 
