1899] VARIATION-STATISTICS IN ZOOLOGY 331 
character may be deduced for any assumed number of individuals ; 
since a curve of variation is a curve of probability, the range depends 
really upon this number; for instance, a variant of the probability 
rod 18 hardly to be expected among only 100 individuals. On the 
contrary, if we find with an otherwise harmonious curve of variation 
some single extreme variant empirically more abundant than it ought 
to be according to its probability, we may conclude that this variant 
did not arise by normal variation, or at least not exclusively by it, but 
that it has been produced by pathological conditions. This conclusion 
is to be controlled by determination of the correlation-coefficients 
which we shall discuss later on. Thus my attention was directed to 
the hitherto apparently unknown ability of Syngnathidae which have 
lost the posterior segments of the body, to regenerate not only a 
complete caudal fin, but probably also a urostyle. I shall refer 
elsewhere to these observations and to some experiments confirming 
them. . 
Comparing several form-units of the same, or of different species, 
as to a single numerical character, all possible differences of the latter 
must clearly be recognised in the differences of its four statistical data. 
Having investigated all the form-units of a species in respect to 
a single character, we should find by graphically representing the 
results a system of curves of variation partly overlapping, of which 
the centroid verticals would be more or less distant from each other, 
while the indices of variability would be nearly constant. One set of 
the form-units represents the constitutional differences of the species 
due to sex and degree of development, the others correspond to 
differences in the external conditions. If the latter conditions have 
influenced not only one character but several characters at once, the 
species has been split up into races or varieties. 
Up to this point we have dealt only with variation of a single 
character within the form-unit. But since all the characters vary, we 
must investigate whether they vary independently of each other, or 
whether there is possibly any relation between the variations of 
different characters. Here also we have recourse to calculating 
probabilities. Every one knows that the probability of the coincidence 
of several events independent of each other equals the product of the 
probabilities of each single event. From each deviation from this 
condition within a larger series of observations we may conclude the 
existence of some causal relation between the events, that is in our 
case between the individually combined variants. This causal relation 
can be a direct one, if the variants of one character are causes of the 
other (correlation sensw stricto) or an indirect one, if both characters 
depend upon the same causes of variation (symplasy). Thus, by 
simply comparing the real with the probable frequencies of the 
individual combinations of the variants of two characters within a 
larger number of individuals of the same form-unit, we are always 
