PROPERTIES OF CLASSES OF FUNCTIONS 19 



is a system of functions 5""' on ^ for which there exists a representative 

 system dt^ = ((r"*')) of the development A such that 



(la) e :^:3me}{m'^me.p) .3. A{Y,57-1) ^e . AiJ^Ab;"-!) ^e; 



(16) ju . 3 . 3 Mo 3 (e : 3 : 3 »;„,. 3 (m ^ 7nle • p) ■ => . 



T.An^.a;" SeA„J. 

 k 



Here the indices g, h, are understood to be g", h"; and the condi- 

 tions (la), (16) are understood to apply only in so far as such indices 

 are available, that is, for the various stages vi the condition (la) 

 applies only for those elements p which belong to at least one class of 

 stage m, and the condition (16) applies only for those elements p 

 which belong not to every existent class of stage vi. The condition 

 (la) is an absolute condition, and (16) is relative to the class 3DJ. 



The system !D(2)t) need not actually depend upon the class M. 

 A system 3) : 



(10 3) - ((S-O), 



satisfying (la) and 



(1'6) m.p .h" .^. 5;'"" = 0, 



is a system ®(9D^) as to every class 9}?. Such are most of the de- 

 velopmental systems given later in this paper.* 



We notice that in case the development A is vacuous at stage m 

 then both (la) and (16) are holding vacuously for that stage, i. e., 

 for no p is there an available index gp and there are no representative 

 elements r'"*. 



18. t The property A of classes 9)J of systems (3t; '^■, A; 9)J).t — A 

 class 502 of a system (31; 'iP; A; 9J?) is said to have the property A in 

 case there exists a developmental system 3) (3)?) belonging to W. In 

 symbols 



9jj^ . =. 9w 9 (3 ©(w) » ^my""")- 



18a. Propositions. — 



(1) 9[R^. = .(3l2rj)\ 



(2) 90^'^ 3 . im/'^. 

 (3)§ 9[>?^''^ . =. . TO/-^ . 



*§48. 



tl. G. A, §79. 



t We speak of a class ^, a development A of ^, a class Ti of functions on 5}S to 3(, as a 

 system (31; '?; A; iUi). 



§ In I. G. A., § 79 (3) is given with D instead of Di, but Di is all that is necessary. 



