404 KANSAS UNIVERSITY SCIENCE BULLETIN. 



of type w in the present set of symbols a new set of 

 the same type as that in § 22. And all new series 

 being fitted in by the nature of the process between 

 consecutive elements of the series previously defined 

 we have at every stage a totality of symbols forming a 

 series. 



89. The group properties of the two lowest rules 

 are preserved throughout the preceding process. On 

 every new symbol not obtained from the preceding 

 by postulating them is built up a set forming abelian 

 semigroups on both the lower and lowest rules, the new 

 symbol playing the part of a unit precisely like w in 

 the series of § 22, and in this set the lower rule is dis- 

 tributive over the lowest. This process is to be con- 

 tinued as long as new symbols can be obtained by it. 



90. There remains one more possibility of building 

 new symbols by introducing a combination defined one 

 way of the higher symbols with those of the series 

 K [2^ o] according to the lowest rule. Thus to every a 

 in the higher series will be assigned a ^(;-series a , slou, 



2iOU' , 2iOU" , diOU'"^ . . . 



This series fits in between a and its next following 

 element in the former set so that we still have a to- 

 tality forming a series. 



91. For completeness it is convenient also to define 

 combinations one way of certain symbols with those of 

 K ['i^o] according to the other two rules, viz.: those 

 expressions in which the symbols of K[i^o] have 

 already been used as marks or indices, e. g. : 



u'w, u"Wy u'"w, . . . W""' , W" , W'" , . . . 

 u'w\ u"w' , u"'w' , . . . w''^' w'""" , w'"^'" , . . . 

 Expressions uoa, u' oa, . . . u'"" , u""", . . . 

 u'^^'y u"""' , . . . are not defined, neither are there 

 any combinations according to the higher rule except 

 the set w" where a belongs to this set. The higher 



