A CLASSIFICATION OF GROUPS 307 



the smaller groups; and that it is not only possible to find one 

 combination of coefficients to satisfy any of the larger groups 

 but frequently very many, the coefficients being so intimately 

 related numerically. 



To discuss thoroughly one of the above groups, consider the 

 240. Before it was beat by observer S, he beat the two groups 

 39 and 54; then he beat the 240 and after it the 15. Now the 

 sum of two 39 groups and three 54 groups equals 240, a number 

 which agrees precisely with the coefficient of the 240 group, 

 and from this it might be concluded (and under conditions prop- 

 erly though not necessarily so) that the 240 group is actually a 

 succession of 39 and 54 groups each the requisite number of 

 times. And the relation between the 54 and 39 groups itself 

 would strengthen this conclusion for, together with the 15 group 

 which the observer beat immediately after the 240, the 39 equals 

 the 54 (39 + 15 = 54) indicating a relationship between these 

 two groups which enter apparently into combination to form the 

 240, the more so since 240 is also a multiple of 15. 



Consider also the 154 group: it is evidently a multiple of the 

 22 group beat immediately before i.e., 7 X 22 = 154; but fol- 

 lowing it are the groups 57 and 20 which in the combination 

 57 + 20 + 57 + 20 also equals 154. A question of consider- 

 able significance arises, for it may at once be asked how it is 

 possible for a group to have more than one set of components, 

 particularly where there is no apparent relation between the 

 several members of the two or more sets? Thus, while seven 

 times 22 is equal to 154, it is difficult to see how this same 22 

 group can be a component seven times of the groups 57 and 

 20 which are also components of the 154 groups. Without enter- 

 ing into a discussion of this question at this time, it may be said 

 that it is for this reason that the writer does not expect to find 

 his analytical results objectified in the records themselves, for, 

 of the many analytical possibilities, which ought one neces- 

 sarily find? are some of these possibilities actualities and others 

 not? and if so, what criteria are to be used in establishing the 

 identity of the right combination? 



