A CLASSIFICATION OF GROUPS 285 



In the light of the above the general series, 



(5) a-G, b-G, c-G, d-G, e-G, w-G, x-G, y-G, z-G 



can be transformed into the series, 



(6) a-G, b-G, a-G, b-G, a-G, w-G, x-G, y-G, z-G 



in which the coefficients a, 6, etc., represent groups of the order 

 of the 22, 46, 70, and 94 of observer R, and the coefficients 

 w, x, y, and z represent groups that are for the present insignifi- 

 cant on the basis of the frequency criterion. The stability of the 

 groups 22, 46, 70, and 94 is established by virtue of the possi- 

 bility of the above transformation. 



2. The coefficients a, 6, c, etc. The coefficients a, b, c, d, etc., 

 of general series (5) have been shown to be recurrent or stable 

 in the previous section. Substituting for a, &, etc., in (6) the 

 particular values as found for observer R by the frequency 

 criterion, 



a = 22; b = 46; c = 70; d = 94. 



The facts are these: a certain individual R beats 17 series of 

 groups on seventeen consecutive days, and of the 303 groups 

 beat, amongst which there are 69 different groups, he beats four 

 groups, the 22, 46, 70, and 94 relatively very frequently. The 

 question to be asked is this; have the abnormally high frequencies 

 of the four groups 22, 46, 70, and 94 four causes or less than four 

 causes? 



A glance at graph 1 will reveal a certain regularity or even 

 periodicity in the maxima for the groups 22, 46, 70, and 94. It 

 is true that the maxima for these several groups do not have 

 even approximately the same values, which is explicable by the 

 discussion of a previous section and also by certain facts about 

 to be discussed. To the apparent periodicity of the above four 

 maxima, the following other singularities may be added: consid- 

 ered day by day (table 2) these four groups do not have their 

 maxima on the same days. Consider the 22 group ; for whatever 

 reason, observer R beats the 22 group with increasing frequency 



