1893.] Correlated Variations in Carcinus moenas. 325 



2. The Correlation of Pairs of Organs : Gallon's Function. 



The method adopted to determine the degree of correlation between 

 two organs was that proposed by Mr. Galton (' Roy. Soc. Proc.,' 

 vol. 40, p. 63). The measures obtained were sorted into groups, such 

 that in each group the deviation X of an organ A from its average 

 was constant. The mean deviation from its average of a second 

 organ B was determined in each of these groups. Calling y m the 

 mean deviation of B, the ratio y, tt /X was found to be approximately the 

 same for all values of X. The same individuals were then sorted 

 into groups in each of which the deviation Y of the organ B was 

 constant : and in each of these groups, x m , the mean deviation of A, 

 was determined. As before, the ratio x m /Y was approximately con- 

 stant. 



Mr. Galton has shown that if Q a , Qj be the probable errors of the 

 organs A and B respectively, then 



=,, a constant. 



The constant here denoted by r is evidently a measure of the 

 degree to which abnormality in one organ is accompanied by ab- 

 normality in a second. It becomes + 1 when a change in one organ 

 involves an equal change in the other, and when the two organs 

 are quite independent. The importance of this constant in all 

 attempts to deal with the problems of animal variation was first 

 pointed out by Mr. Galton in* the paper already referred to : and I 

 would suggest that the constant whose changes he has investigated, 

 and whose importance he has indicated, may fitly be known as 

 " Galton's function." 



As an example of the mode of determining this function, the cor- 

 relation between the right and left antero-lateral margins of the 

 Naples crabs may be examined, by means of the data given in the 

 following tables. 



In Table IV the measurements of the right antero-lateral 

 margin have been sorted into groups, each group containing 

 individuals which differ by not more than 0*004. of the carapace 

 length, the magnitudes included in each group being given in the 

 first column. In each of these groups the mean size of the left 

 antero-lateral margin was determined, and the value obtained is 

 given in the second column. From these data, and from the re- 

 ciprocal data of the next table, the mean value of Galton's function 

 was found to be 0*76, and the extent to which this value fits the 

 individual cases may be estimated from the third and fourth columns 

 of the table. In the third column, the value of the left antero- 

 posterior margin corresponding to every value of the right margin 



