298 CARL W. BOCK 



118 2 times, 333 (334-1) 1 time, 



142 1 time, 416 1 time. 



190 1 time, 



It is hardly necessary to comment on the remarkable accuracy 

 of the above predictions and it is believed that the concurrence 

 that obtains between prediction and fact is sufficiently great to 

 warrant the belief that the groups 118, 142, 190, 333, and 416 are 

 of the same order as the 46, 70, and 94, and therefore of the 

 nature of the 22 group. It should be pointed out however that 

 frequency criteria cannot be invoked in the cases of these groups 

 to establish the said relations partly because they are very large 

 groups and partly also for reasons later to be considered. Since 

 the general series suffers no radical modification by the above 

 discussion, no transformations will be made; the general co- 

 efficient x may be taken to stand for all multiples of the type 

 group a that can be predicted as in the above. 



5. The coefficient a. With the exception of the general co- 

 efficients y and 2, all other coefficients have been shown to be 

 definable in terms of the coefficient a. It is interesting to ex- 

 amine whether a can be expressed in terms other than itself. 

 A rigorous application of the mathematical law of no exception 

 would lead to the conclusion that such groups as the 22 might be 

 analyzed and expressed in terms of groups smaller than itself in 

 much the same manner as the groups 46, 70, and 94 have been 

 expressed in terms of the 22 group. 



The application of the above principle however carries with it 

 the necessity for giving to its results a very clear meaning which 

 may not be simply conventional or arbitrary, but which must have 

 some relation to, and which will square with, the morphology 

 and physiology of the organism which has produced the groups in 

 question. In consequence thereof analyses may be carried out 

 only so far as they are possible of interpretation, the limits and 

 meanings of which we are about to examine. 



The facts or data on which an analysis of the 22 group might 

 be possible are these: the hypothetical components must neces- 

 sarily be smaller than the 22 group; of the smaller groups which 



