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, 

 Siou', 2ion" , Siou'", . . . 



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['Zio] according to the other two rules, viz.: those 

 expressions in which the symbols of Kfiio] have 

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



u'w, u"w, u"'tv, . . . w' , W"^" , W'" , . . . 

 u'w' , u"w' , u"'w' , . . . w'""' w'""" , w'""'" , . . . 



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

 u'""', 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 



