40 VARIATION AND CORRELATION IN THE CRAYFISH. 



INDEX CORRELATIONS. 



It has been elsewhere pointed out by the writer of this paper (Pearl 1906 

 and 1907) that it is of considerable theoretical importance to determine for 

 as many organisms as possible whether there is or is not a sensible correla- 

 tion between the proportionality of the parts in a differentiated system of 

 organs and the absolute size of the system. An essential part of Driesch's 

 so-called "first proof of the autonomy of vital phenomena" (cf. Driesch 

 1901) depends on the assumption that such a correlation does not exist, 

 but that, on the other hand, proportionality of parts and absolute size are 

 quite independent in the organism. Fortunately the matter is one which 

 can be quantitatively tested and a definite answer reached by direct appeal 

 to the facts. The proportionality of a series of parts can always be measured 

 by forming from the absolute dimensions of these parts a series of indices 

 which will give the percentage which the size of a given part is of the 

 size of some other part or of some dimension measuring the size of the 

 whole organism. Having determined these indices, in order to answer 

 our question we have merely to calculate by now well-known mathematical 

 methods the correlation between a given index and any chosen measure 

 of the absolute size of the organism. The mathematical theory of index 

 correlations was first investigated by Pearson (1897) and Galton (1897). 

 A discussion and illustration of the meaning of the formulae has been 

 given by the present writer in another paper (Pearl 1907) and need not 

 be repeated here. We need merely to note that if x l /x s be any index, 

 then the gross correlation between this index and the absolute dimension 



a? 3 is given by the expression 



ri 3 vi v 3 



P = -~ 



1/Vi 2 + v 3 2 2ria Vi v 3 



and the spurious correlation between the two characters which exists 

 when all organic correlation between x } and a? 3 is destroyed is given by 



_ 



" 



where r n is the coefficient of correlation between x l and x 3t and v 1 and v f 

 are the coefficients of variation of these two characters. 



In the crayfish appendages we have a system of parts in which definite 

 proportions are maintained with a very high degree of constancy. This 

 is an obvious fact from even cursory observation, and quantitative proof 

 of it is given in table 6 above. It seems an especially suitable object on 

 which to test our question as to whether or not these proportions are cor- 

 related with the absolute size of the body. To get at the question in a 



