332 GEORG DUNCKER [November 



able to find out whether there is correlation between these characters 

 or not. 



For numerical characters there are now simple methods (Galton, 

 Pearson) of calculating the degree of deviation of the real frequencies of 

 the combinations of their variants from the probable ones. The results 

 of these calculations are abstract numbers between zero (no deviation 

 from probability) and one ; the latter signifies the highest possible 

 degree of deviation of the combination-frequencies from probability, 

 inasmuch as each variant of the one character occurs only combined 

 with a definite single variant of the other. These abstract numbers 

 we call the coefficients of correlation of the investigated pairs of 

 characters. The most convenient coefficient of correlation is that 

 calculated according to Pearson's method [13], and determined as the 

 mean product of the individually combined relative deviations of the 

 two characters from their average values, while the relative deviations 

 are the absolute ones expressed in terms of their indices of variability. 

 Like the indices of variability of homologous characters, the coefficients 

 of correlation of homologous pairs of characters show a certain 

 constancy even in different species (Warren [17]). This, again, I 

 believe to be an expression of physiological relations between the 

 correlated organs with regard to the respective characters. 



If on the average the combined variants lie either, on the one 

 hand, both above or both below the mean values of the two characters, 

 or, on the other hand, if the one is a positive, the other a negative 

 deviation from these values, we get either a positive or a negative 

 coefficient of correlation, and accordingly deal with positive or negative 

 correlation. Series of variation between which there is positive 

 correlation tend to form constant differences of the individually com- 

 bined variants, while those between which there is negative correlation 

 tend to form constant sums of the variants. The constancy of these 

 sums or differences is the more remarkable, the higher the coefficient 

 of correlation. The constancy of the sums of variants, that is, negative 

 correlation, is mostly to be found in metamerically disposed characters 

 (homoiotic variation), that of the differences of variants, that is, positive 

 correlation, in antimerically disposed characters, especially in those with 

 symmetrical variation. 



As it is possible to investigate the probability or the degree of 

 correlation of the frequencies of the individual combinations of the 

 variants of two or more characters, so it is reciprocally possible to treat 

 the combinations of variants of one and the same character in two or 

 more individuals which are connected by known relations in a similar 

 manner. This might be, for instance, when we wish to decide if a 

 character is important in sexual selection ; or again, if a character is 

 hereditary or not. In the former case the combination of variants, 

 effected by mating, of one and the same character in males and females, 

 must show correlation ; in the latter the same is true for parent and 



