28 pitcher: interrelations of 



(2) follows from §23all, and (1). (3) follows from (2) and the con- 

 sideration of the class (9!}?'9Jr')* in case W = [0]. (6) follows from 

 §23alli and the following proposition: 



(7) 9jr 9 m' . 3 . Mv = m;, : 3 : e""'": z> . d',., = e;,,. 



This last proposition gives rise to a more general one: 



(8) m'^in' .z^. n',,. = m;-) . m" ■. d : e^"'"'^""'- . ^ . e,.,„ = e,.,.. (p"). 



27. Propositions. — 



(1) A^':3:W^'.~.TO^X 



(2) A^':D:9Jf^'=.~.95J'f'=. 



(3) A''^:z>:.3)l''\^.m'''°: 



(4) A''=.9[)J'^'.z).5[rj = [0]. <? 



(5) A''^93Z:3:g«^-.~.g)J'^'-.cx..9« = [0]. 



(6) A'-"^' : 3 : ^'^vivot'o _ ^ _ ^K,,'mm">_ 



(7) A'^' : =) : ^''^"-"'^'■''"> . ^ . ^^.="»vk";_ 



(8) A''" : 3 : ^'^■''^'e-""' . ex. . ^«:.='»'ci'i")_ 



(9) A' '■= : 3 : ^'^.'■="'*"'"' . 3 . ^p-p. = ( p'p") . 



(10) A^' : 3 : a}J^" . ~ . »?^'=-. 



(11) A''':3:5m^-.~. 9[«f:-. 



(12) A'':3:9)J^'".~.9[)Jf'=-. 



(13) A.a«^ = [0].3. A'\ 



In (1 . . . 5), (10 . . . 13) 2)? is a class of functions on ^ with 

 a development A conditioned as indicated. In (6 . . . 9) ^ is a func- 

 tion on "^ = f^", y}l' and W are classes of functions on <p', ^V re- 

 spectively, and A' is a development of '^' conditioned as indicated. 

 (1, 2, 3) follow from propositions (1, 2, 3) of §20a, and (6, 7, 8) 

 follow in a like manner from (1', 2', 3') of the same section. (5) fol- 

 lows from (4) and from §23alli. (10, 11, 12) follow from (6, 7, 8) 

 and the definitions of the properties K[^ K'^^ -K^i2*- Thus the proof of 

 (10) is as follows: 



