22 



pitcher: interrelations of 



respectively in case each function ju of SDJ has the property 



A'lTO; XsTO; K,m. 

 In symbols: 



(9) 50J*'': = :iO?^'''-'': = :/x.3.M*''™; 



(10) W°- : ^ : W'"" : = : M . 3 . m^'=" ; 



(11) 9!)?*'=: ^ : W"'"':^ : m • ^ . m*'=™. 



The properties Ki; Ko; Ku are properties of special reference in 

 that they refer to a development A of the class ^. 



Relative to the ordinary developments* of ^'; ^""; ^'"; '=p''', 



the classes W; 9JJ"-; m' 

 Ki, Ki, Ki2- 



20a. Propositions. — 



(l)t . 



(2) • 



(3) 



I. grij.n.. 



A^' . m 



A^' . m 



9DI'"'; 50?'^ possess the properties 





In (1, 2, 3) the properties Ki; K2 are said to hold vacuously, viz., 

 in (1) there exists a stage ?reo such that for m exceeding Wo there is 

 no element p satisfying the relation A'p„; and in (2) there exists an 

 Too such that for m exceeding Too no two elements pipi satisfy the re- 

 lation Kj,^p^n- The property Fi is, by § 16a6, a special case of Fi. 



Similarly if v? is a function on ^"^" and W a class of functions 

 on 'ip' and W a class of functions on ^" we have the propositions: 



(1') ^'''\<p.w .m" .z^. ^^v-w"'; 



(20 A''"' .<p.m'. m" .z>. <p 



(30 A''\<p.W .W .^. <p" 



21. Composition of developmentst A. — Consider two developments: 

 A'-((^""")); A" 

 with corresponding sequences: 



AVWi.'St"). 



i'o'Bi'(3)i") 



((^"""")), 



{I'm); 



n' 



1 I'v 



§15. 



t Every class 331 of functions has the property Ki; K^; Ki, with respect to an ulti- 

 mately complete; ultimately fine; ultimately vacuous development A. 

 1 1. G. A., § 83. 



