20 pitcher: interrelations of 



19. Relations K between the elements of a class ^ icith respect to a 

 development A. — We define at once what we mean by the symbols* 



Kp„, . =. (pm) a 3 mo ^ m s p'""; 

 Kp,p,^ . =. {piP2m) 9 3 (Too ^ TO . h ^ L,) 5 (pr'" . pr'» )• 



that is, 



Kp,„ denotes an element p for some stage TOq ^ to not developed; 

 Kp,p,^ denotes two elements pip2 (not necessarily distinct) belonging 

 to the same class of some stage Too ^ to. 



Relative to classes *iV; <}?"; etc., we denote relations above as 



In the ordinary developmentf of ^X^'" the relation Kp„, is equivalent 

 to the relation p > m and in the ordinary development of 1^'^ the 

 relation Kp^p^^ is equivalent to the relation A{pi — j)^) < 1/to. 



20. PropertiesX Ki, K-2, K^ of classes ^l of functions of systems 

 (2t; ^; A; m).—We define directly the properties KiW; KM; KiM 

 of <p, where ^ is a function and 502 a class of functions on '1^ to 2t. 

 The definitions are relative to a development A of ^].^. 



(1) (p'"''"-^. = . (p3 (3fji9e -.^ -.3 nie i Kpm^ .zi. Aipp ^ eAfip), 



viz., the function ^ has the property KM in case <p is such that there 

 exists a function n of 9.1J such that for every positive number e there 

 exists a stage to^ of the development such that the relation 



implies 



A<pp ^ eAfXp. 



(2) ,f'^'-'" .=. ^9{3fx9e : z) : 3 m.^Kp^p^m, ■ 3 . A(s?,„ - <Pp,) ^ eAup^), 



viz., the function ts has the property K^'iffl in case (p is such that there 

 exists a function fi of SIJJ such that for every positive number e there 

 exists a stage to. of the development such that the relation 



* The relations here defined are the relations Kf, A'.f of I. G. A., §77. In §69 and 

 following of the work mentioned Moore discusses general K relations of which the relations 

 K-^ are special instances. 



t Cf. § 1.5. 



t The properties here defined are the properties K^, Kf, A'f^ of I. G. A., §77, and 

 are special cases of the more general K properties discussed by Moore. 



