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lections, provided that they are distributed over their common 

 area uninfluenced by causes tending to bring them together 

 into the same situations, and if the actual avei'age of the joint 

 occurrences of the species is one in every five collections, then 

 the associative bond of the species concerned may be given 

 the value of four — a value of little signiflcance perhaps, taken 

 by itself, but useful, at any rate, for a comparison of the darters 

 with otiier groups. And if certain of the species are associated 

 with the other darters in an average ratio of five to one, while 

 other species are associated with the other darters in an aver- 

 age ratio of only two to one, then the former species will typify 

 the ecological group more definitely and correctly than the 

 latter. 



By this means, also, if the actual frequencies of joint occur- 

 rence of the various species of the group be compared with the 

 computed average of such frequencies, the division of any pre- 

 sumably single group into two distinguishably separate ones 

 might be made out. If it should appear, for example, that the 

 species of darters may be divided into two groups, each of 

 which taken separately is found to have a mutual associa- 

 tive ratio of six to one, while the corresponding ratio between 

 the two groups themselves is but three to one, we may infer 

 provisionally the division of the darters into two ecological 

 groups, distinguishable by their predominant attraction to dif- 

 ferent sets of ecological factors in their common environment, 

 but united in turn in one larger group by their common at- 

 traction to certain other factors. 



For an analysis of the facts, we need for each species of 

 darter a determination of the average frequency of its merely 

 chance occurrence in collections with each of the other species, 

 a determination of the actual frequency of these joint occur- 

 rences, and a numerical expression of the ratio of one of these 

 frequencies to the other. Then by a systematic tabulation of 

 these latter i-atios, which may be called the coe(firie)its of associ- 

 atioH. we may compare one species with another, and bring the 

 essential data for the whole family under the eye for conve- 

 nient inspection and analysis. 



