A CLASSIFICATION OF GROUPS 297 



Accordingly groups like the 44, 45, 47, and 48 will, be consid- 

 ered as varieties of the group species 46, and it will be predicted 

 that in any similar investigation such varieties will occur. The 

 coefficient w thereby suffers the following transformations: 



Wi = a =1= n, w 2 = (2a + k) n; w s = (3a + 2k) =t n; 

 W4 = (na + (n l) k) =*= n. 



4. The coefficient x. The general series (6) contains the un- 

 defined coefficient x whose nature it is the purpose of this sec- 

 tion to discover. The simple facts upon which the discussion is 

 based are as follows: an organism beats a certain group a very 

 often; it was also shown that he beats groups of the order 2a, 

 3a, 4a, etc. The question to be asked is whether any other 

 multiples of a might be expected to occur which cannot be dem- 

 onstrated to be such by an appeal to simple frequency criteria? 



The following numerical equations obtain between those 

 groups that have been shown to be multiples of the group 22: 



22 = 22 



46 = 22 + 22 + 2 

 70 = 46 + 22 + 2 

 94 = 70 + 22 + 2 



Continuing these equations with their numerical implications, 

 the following expressions can be obtained: 



118 =94 + 22 + 2 262 = 238 + 22 + 2 



142 = 118 + 22 + 2 286 = 262 + 22 + 2 



166 = 142 + 22 + 2 310 = 286 + 22 + 2 



190 = 166 + 22 + 2 334 = 310 + 22 + 2 



214 = 190 + 22 + 2 358 = 334 + 22 + 2 



238 = 214 + 22 + 2 382 = 358 + 22 + 2 



416 = 382 + 22 + 2 



In the above equations the numbers on the left hand side repre- 

 sent groups that might be expected to be beat by this observer 

 in view of the fact that he beats the 46, 70, and 94 groups. Of 

 these, the following groups are beat with the indicated fre- 

 quency by this observer: 



