398 KANSAS UNIVERSITY SCIENCE BULLETIN. 



71. In every one of the four cases v^hen 



Wi has a first element or V^ has a last 

 Vz has a last element or W^ a first 



this element is called the limit of the set to v^hich it 



belongs. 



72. In case W^ and V^ or V2 and W2 are both un- 

 bounded, v^e v^ill introduce a nev7 symbol f^ v^hich we 

 v^rillcallthelimitof K[fJ. 



73. Lemma. If Wi and Vi are both unbounded for 

 the series K[i^] the same is true for the series 

 K[aof^] v^here a is any symbol in S. For if either 

 the W or y set corresponding to K [a o f „] had a limit 

 I we could v^rite l = aox and x would be a limit for the 

 same W or V belonging to K [f ^] contrary to hy- 

 pothesis. 



74. Theorem. If the series K [ f J , K [g,] both 

 divide S into two unbounded sets the same is true of 

 the set K [f ^ og^]. 



Proof. Lethi = fiOgi, ha^fgOga, . . . . 

 h^+i^fgogi, h^+g^fgogg, . . . 



Then by the Lemma each of the series K[hJ, 

 K[h^^,], K[h,^^,], . . . K[h,^^.,], . . . divides S 

 into two unbounded sets. Hence the same is true of 

 tnesetni, 11^+2* ^2w+3) • • • '^iw+i+i- 



75. On the basis of the preceding theorem and defi- 

 nitions is built up, by laying down the usual definitions 

 of equality, order and two rules of combination, the 

 set X of limits of series K[f„] selected out of R. The 

 set X forms an abelian group with reference to the 

 higher rule of combination and an abelian semigroup 

 on the lower rule. The higher rule is distributive over 

 the lower and possesses the modulus u. X may be 

 simply ordered according to the lower rule and is di- 

 vided by the modulus u into two unbounded sets each 



