K. Pearson 
207 
merely another way of looking at the change from discrete to contimions variation, 
due to the influence of a multitude of causes on the discreteness of the variates 
which fall into a given Spielraum. 1 still find nothing absurd in the statement 
that the actual effect of the scatter is sensibly equivalent to a fractionising of the 
indices. It is simply equivalent to the statements, (i) that the ordinates of the 
Gaussian curve closely give, even for small values of n, the terms of the binomial, 
(ii) that the ordinate of the Gaussian curve between two terms of the binomial 
closely gives a fractionised binomial term (owing to Stirling's theorem being true 
for fractionised factorials or V functions), (iii) that we have no knowledge of how 
the "scatter" within the Sjnelraiim may be distributed so as to give a continuous 
effect*. Now these points are not in the least needfid for the deduction of my 
skew curves, they are merely given here because in our complete ignorance of the 
nature of the causes, hereditary and environmental, which produce continuous 
variation, I think we have no warranty for saying that a limited number of cause- 
groups is impossible, or that no such limited number of fundamental cause-groups 
could give a continuous variation. In the present state of our knowledge we 
cannot agree with Ranke in sweeping away as impossible all the discreteness which 
follows from determinantal theories of inheritance. We cannot afford to be 
dogmatic as to the continuous or discontinuous character of the ultimate sources 
of variation and any effective theory must like the Laplace probability integral be 
equally applicable to the sum of a discontinuous series as well as to the areas of 
a continuous curve. 
(b) Any finite series of cause-groups, Ranke tells us, must lead to dis- 
continuity. 
I have endeavoured to show above that the discontinuity may be as real and yet 
as undetectable as the distribution of lengths, say, of the vertebral columns of sharks 
which yet depends on the number of discrete vertebrae, with a scatter of their 
individual sizes. But Ranke's argument in itself is a false one, many discontinuous 
systems lead at once to continuous distributions. In our ignorance of the exact 
sources of variation, all we can do is to show that a limited number of cause-groups 
can quite well lead to continuous variation. To take a perfectly arbitrary 
illustration, suppose that a character can only take values lying between «i and cu 
and that this character is to be settled by the determinants derived from s + l 
ancestors, i.e. suppose all but s + l to be cast out in the successive divisions of the 
germ-cell. Then it by no means follows that the character will be a blend of these 
s+l determinants, one or other of them may be dominant. It does not follow that 
the dominant one represents either the one with the least or the greatest value of 
the character. It might be the one with s — r determinants below and r above it. 
For example I have a variety of Binomial machines or "Quicunxes" like that figured in my 
memoir, Phil. Trans. Vol. 186 A, Plate i. Fig. 2, p. 414. It is quite possible to arrange a quicunx in 
which there are only a limited number of compartments, but in which the top of the seed in these 
compartments is not horizontal, bnt gives a continuous curve, e.g. the greater air draft of the greater 
frequency might be used to pile up the material in any receptacle on the side of the greater frequency. 
