A CLASSIFICATION OF GROUPS 303 



The question to be answered is this: what relations exist be- 

 tween any of the above short series of groups and the corre- 

 sponding numerical equations? It must be noticed first that all 

 the numbers which occur in equations (1) to (13) occur also in 

 that part of the series to which they correspond, and that no 

 other number which does not occur in the corresponding part 

 of the series occurs in the equations. Two possible conclusions 

 can be drawn from these correspondences; either we are con- 

 cerned with a remarkable set of coincidences or the above nu- 

 merical relations have some significance. The first conclusion is 

 untenable from the fact that such numerical relations as be- 

 tween the coefficients of groups are by far too common to be 

 regarded as accidents, and secondly from the fact that in prin- 

 ciple the above equations do not differ from similar equations 

 obtained in the case of observer R except that in the latter 

 case the equations involved the addition of identical coefficients 

 whereas here they involve the addition of non-identical coeffi- 

 cients. From this it is obvious that the general coefficient y 

 may be taken to represent groups whose coefficients are ob- 

 tainable by the addition of two or more different groups which 

 latter groups must occur in the temporal vicinity of the groups 

 whose components they may be said to be. 



An extension of the arithmetical principles involved in the 

 above equations is possible when they are obtained not only as 

 between neighboring groups of the same record, but also be- 

 tween any groups at all within the said record. Below are given 

 in categorical form a list of the more important equations ob- 

 tainable from among the several groups of the series under dis- 

 cussion. With certain exceptions, the equations do not contain 

 any numbers that are not found as such among the coefficients 

 of the groups of the above series. The exception mentioned 

 above consists of the use of the half of the coefficient of groups 

 in the case of even numbered groups, and approximate halves 

 or so-called physiological halves of odd numbered groups. The 

 only justification for the use of said numbers in the making of 

 analytical equations as above comes from the fact that it is 

 so generally possible to do so, and from the obvious relation that 



