40 pitcher: interrelations of 



We notice that A^" implies the existence of a stage m^ such that 

 for any succeeding stage m the class '51?'" contains only a finite number of 

 elements and the class u[^?'"j?n > m^] is denuniberable. Recalling 

 § 34 we see that propositions 4, 5, 6, of the present section and Theorem 

 I are to be applied especially to 'ip"'. 



39. Notations and definitions. — In § 14 are given the definitions 

 of the classes *}?'" 'iP"'", '^m, etc. In this section we define other sub- 

 classes of the class '^, specify notations, and define certain properties 

 of functions and classes of functions. These definitions and notations 

 are to be used in the immediate sequel. The first two definitions are 

 relative to a class "ip, the others relative to a class '^ with a develop- 

 ment A. 

 (1)* 3)l^K^. m?{3{ix.e)9p.^.Afx,^ e). 



(2)t ^^K^.,xi{3e,p.z,.At,,-^e). 



(3)t V-„..^.[p-^]^%^-V\.. 



(4) A%„.=.(/j, m)jp^»^". 



(5) Xp..=.(p, «0 9p^»»-". 



(6)§ Kj,„,^m,-=. (p, mi, m.,) s (3pi 3 A'p,„„ . Kpf\r,l^. 



\> ) -Pin I'M • ^" VP-mivl^ • ^' [3'li P ' -t^jimimj. 



^m, ^-,n, "Pmj^j may be considered as reductions|| of "]?. We speak 



of the corresponding reduced developments^ l^s„i ^r.„, ^r„.„, and of 

 the reduced functions** /xj;^, m/; „. y-n^^, and classes** of reduced func- 

 tions 3)?^,„, ^\n.^, Mn^^^,- 



* 5DJ is said to have the property Bz in case 9J! is such that there exists a function ii 

 of 9}}, uniformly bounded from 0. 



t (2) is a special case of (1) where the class 3R consists of a single function m- 



I The class ^_m is the class of all elements each developed at every stage ??(o = '«. 



§ p, ?«!, jHo are in the relation Kpm^m„ in case there exists an element pi developed at 

 every stage ?«o = mi, {Kp^„^, such that p is directly connected with pi at some stage 

 m'^m., (A'ppj^j). 



II §11. 

 If §22. 

 **§11. 



ft 9K has the property Bz in case there exists a function ^ of fflt such that for every m,/i 

 is uniformly bounded from on ^_m. 



