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 

 TotJo~o i s 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 sensu 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 



