279 



The computation may be facilitated by the use of algebraic 

 symbols. 



Let a equal the total number of collections to be used in the 

 computation; b, the number of collections containing the more 

 abundant of two species to be compared with one another; c, 

 the number of collections containing the less abundant of these 

 species; and d, the number of collections each of which actually 



contains both species together. Then - - expresses the chance 

 that any collection of a will contain one or more representatives 

 of the first species; --, the chance that any collection will con- 

 tain one or more representatives of the second species; — r , the 



chance that any collection will contain one or more represent- 

 atives of both species at once, provided that the distribution of 

 each is ecologically independent of that of the other; and 



be 



— , the probable number of chance occurrences of the first and 



second species together in the number of collections repre- 

 sented by a, the same proviso being made. Since d = the ac- 

 tual number of such joint occurrences, ^— is the formula for 



be 



the ratio of actual to calculated joint occurrences— the for- 

 mula, in other words, for the computation, in all cases, of our 

 coefficients of association. For example, substituting in this 

 formula the values already given for Hadropterus aspro and 

 Had) 'optei us ph oxoceph alus, 



ad 1000 X 40 2 96 



be ~ 159 X 85 ' ' ' 



To determine the coefficient for any pair of species, we 

 need only to know their separate frequencies and their joint 

 frequencies in collections derived from the territory of their 

 common distribution. 



The above formula may be translated into the following- 

 rule for finding the coefficient of association of any two 



