K, Pearson 
205 
(6) That what I term the "number of cause-groups" must be infinite in 
number, for without such infinity it is impossible to reach continuity. 
(c) That looked at from the standpoint of binomial or hypergeometrical 
series the constants of some one or more of my curves may become unintelligible. 
Now not one of these objections has any application to the method which has 
been used in this paper to deduce the differential e(|uation to my curves. But 
I still think there are very grave objections to every one of the above statements. 
To begin with (o). We meet in an immense variety of living forms with 
discrete variates. For example, the number of teeth on the rostrum of a prawn, 
the number of lips of a medusa, the number of veins in a leaf, the number of 
glands in a swine's foot, the number of tentaculocysts in Ei^liyra, the number of 
individuals in a litter, the number of bands on snails' shells, the number of somites 
in the body of an earthworm, the number of petals or sepals in a flower, etc. etc. 
Are we to put all these distributions of variation on one side because Ranke and 
Greiner hold that all distributions of variates are continuous ? We have in these 
cases probably continuous causes producing discontinuous distributions. Are we 
not to use the areas of a continuous curve to give the frequency of such discrete 
variates ? 
Consider for example the function given by : 
M [ \n \ 1 
0 V ^ |«-r|r V27ro-, ] ' 
This is compounded of n + \ normal curves, the area of the (/•+l)th normal 
curve being Nif'-' q'' J-^^ , i.e. the (/• + l)th term of the binomial Nij) + q)'\ and 
this (r+l)th normal curve has a,, for its standard deviation. The origin of the 
system is at the mode of the normal curve corresponding to r = 0, aud the means 
of these normal curves are spaced equal distances c apart. 
When every a,. — 0 we have discrete variation. When a,, is small, less say than 
^c, it would probably be difficult to distinguish the result from discrete variation. 
Enlarging a,, we pass on till we get a system which it would be practically 
impossible to distinguish from continuous variation, even if n were only moderate 
in magnitude. I lay no stress whatever on the above expression because I am in 
no sense pledged to any Gaussian curve, but it illustrates well what I want to 
express: namely, in actual nature the frequency might fundamentally fall on 
certain values of the character, but that the effect of nurture, environment, and 
growth may well scatter the values of the variable round the fundamental value, 
so that continuity of variation is all that can be actually observed. The number 
of somites in an annulose animal is discrete and probably inherited, but the length 
of the body may appear as a continuous variate. I do not think for a moment 
that the distinction made by Ranke between discrete and continuous variation, 
