PROPERTIES OF CLASSES OF FUNCTIONS 25 



A-pcbb). Thus from § 16 we have the properties F(^]?J, Fi(f5^), 

 V2{V^), Fi^^), F,{%\), F,{^^^), Ca\), C^{%\), C,(^^„) of develop- 

 ments A where ^j^ is any reduction of '-1?. It should be noted that 

 any class ^i of '^ is a reduction of ■"!?. 



We note the following proposition relative to a class •!}? with a 

 development A and a class ^^j^ with a development A^: 



(1) e^V^" . D . ef.^'fl, " {i = 1,2, 12). 



23. The properties* A',^, K^^, K^.^.^ of classes 5D? of functions of 

 systems (21; ^; A; dJl). — In the following definitions W, W are classes 

 of functions on ^', %V' respectively, v? is a function on ''T? = V''V'") 

 and {m'W)^ is the *-extension of the composite class {3)1' W). 



(1) 2)?'*'"'* :^:W 3 2)r '^^"^ . z) . iWm") * = [all ^^V'-^d.") . b,ik'w^ . 



(2) W""-'' : = : 9}r 9 502"^^'' . z> . (WW) =, = [all ^^^'™'<^'") ■ ^«^''V)] ; 



(3) 9JJ'*"'" . = . a«' 9 arj"^^'' . z> . (©r^ro h= = [aii ^^■-'^^'''^'"^ • ^o^'"'^-'] ; 



viz., W has the property K'.^, {i = 1, 2, 12), in case 9[>J' is such that 

 for every W'' " the *-extension of the class WTl" is the same as the 

 class of all functions ^ having the property K\W'{W') and belonging 

 to 2>J" for everj' p' . 



23a. Propositions] concerning relations and classes of functions on '^. 

 The following propositions relative to the class ^ with a general de- 

 velopment A are for i = 1, 2, 12. 



(1,) 9i^'™.3.(A$R)^'^'. (215H)^'^ 



(2,) arjf "-^"^ . 9J^'^'' . 3 . 9t^*™% 



(30 gWf ™= . 9i^-='- . 3 . 9{^'™^ 



(30 ^ at^'''" . (2^'-^=\ 3 . ®*''^ 



(40 3[)?^'.9?^''''.3.9t/'^ 



(50 5m'' . 9{^''" . S^'" • ^•'"" . 3 . 9i%*'^'. 



(6i) gOt'' . 9i*'''" ■ '^''"'' . =) . 5t/'-" • •^'''-' . dl^^''"' ■ •^■*'". 



(70 TO^^' . 3 . 2)?^^^' . Wan^l' . 93?/^^ 



* Cf. I. G. A., § 72. 



t Propositions li-14, are given in 1. G. A., § 72a as relative to A' relations and properties 

 more general than those derived from the development .i. 



