2 pitcher: interrelations of 



functions /j. since its definition is relative to some special feature, e. g., 

 Ii77iil or distance postulated as defined for the range '^. 



The principal properties of classes of functions used in the above 

 memoir are : 



L (linear); C (closed); D (dominance); A (absolute); 

 (1) 



Ll, -fV,, /Vj, iVjo;^, -fVi^, 1^2*} ^v 



These properties are fully defined in the introductory sections of 

 this paper. The first eight are the more important. The first four 

 and the last are properties of general reference. The others are 

 properties of special reference being defined relative to a so-called 

 development A of the class '^. Moore shows that these properties 

 belong to many classes of functions of well-known importance in 

 Analysis. Thus the classes: all numbers; all n-partite numbers; all 

 absolutely convergent series; all sequences converging to 0; all con- 

 tinuous functions on a closed linear interval, each possess the first 

 eight properties. 



In the theory of Moore the development A of the range %^ is of 

 fundamental importance and, as a basis of definition of properties of 

 infinitary or infinitesimal nature like the properties Ki, Ko here in- 

 volved,* is no doubt destined to play an important role in general 

 function theory. In this paper various types of developments of '^ 

 singular (^IV), ^ dual (<|3"=), '!> triple (^13"^,^ finite i^''"),^ denumerable 

 (^'") and ^ general are specified, j All of the independence examples 

 and many of the theorems are in terms of these special developments. 



In the Introduction to a Form of General Analysis the importance 

 of the property iv,,^ is displayed and indeed it is one of the goals of 

 the theory. The possession of this property by a class W of functions 

 insures a certain functional characterization of the so-called *-com- 

 posite class, J (SK'Sf)?")*, where W is an arbitrary class of functions 

 having the properties L, C, D. Such a characterization is of funda- 

 mental importance in the development of the general theory of dif- 

 ferential and integral equations from the standpoint of General 



* Cf. I. G. A., § 67. 



t §§ 15; 16; 47. 



I E. g., if 9J!' and 33!" are, respectively, the classes of all continuous functions of p' 

 and p" where p' and p" have the respective ranges, = p' = 1, = p" = 1; then the 

 ♦-composite class (Sli'3)l")« is the class of all continuous functions of the bipartite variable 

 p'p" which ranges over the plane region: ^ p' ^ 1, ^ p" ^ 1. 



