1918] Michael: Behavior of Sdpa democratica 255 



We have now considered the data in six different ways with similar 

 results. In table 4 data relative to all the June and July hauls are 

 considered, while in tables 5 and 6 night hauls, day hauls, mid-day 

 hauls, morning hauls, and evening hauls are separately considered ; 

 and in each case the following relations are shown : 



1. Solitary forms most abundant in the warmer water. 



2. Solitary forms most frequent in the colder water. 



3. Aggregate forms most abundant in the colder water. 



4. Aggregate forms most frequent in the colder water. 



It must be evident that such a variety in tabulation would introduce 

 contradictions were it not for a definite relation between fluctuations 

 in surface temperature (or some influence closely associated with it) 

 and the behavior of the two generations of this salpa. However, on 

 account of the apparently paradoxical relation between the abundance 

 and frequency of solitary forms, the probability that such a repeti- 

 tion of relations was due to chance, insufficient or inadequate haul- 

 ing, or to anything other than an actual correspondence between tem- 

 perature and distribution has been calculated and found to be less 

 than 0.0007. The calculation is simple, as follows : 



There are four possible ways in which the aggregate forms might have been 

 related to the colder water, namely: 



1. Most abundant and most frequent. 



2. Most abundant and least frequent. 



3. Least abundant and most frequent. 



4. Least abundant and least frequent. 



Whichever of these combinations might result, it is obvious that the solitary 

 forms might have been related to the colder water in any one of the same four 

 ways. For each tabulation, therefore, there would be sixteen possible com- 

 binations of relations, any one of which would be equally likely to occur by 

 chance. Then, for any one of the sixteen possible combinations that appeared 

 in the first tabulation, there would remain sixteen possible combinations that 

 might appear in the second tabulation, whence the number of possible combina- 

 tions in two tables would be 16=. In three the number would be 16^ in four 16^, 

 and in six 16^ Hence, the probability that the same combination of relations 

 appearing in all six tabulations was due to chance is i/ip," or 1 -=- 16,777,216. If 

 tables 4 and 5 as well as section A of table 6 be excluded on the ground that 

 they contain part of the data in the remaining tabulations, the probability would 

 be 1 -■- 16' or 1 -^ 1,5.36 which is less than 0.0007. 



It may be claimed that this is an underestimate because of a natural associa- 

 tion between maximum abundance and maximum frequency. Suppose, then, 

 that the probability of maximum abundance and maximum frequency of the 

 aggregate forms occurring together is large — say P. Then, since there is an 

 equal chance of this combination being related to the colder or the warmer 

 water, the probability of its being related to the colder water in any one 



p 

 tabulation would be — . From this it follows that the probability of minimum 



