30 pitcher: interrelations or 



The following propositions are relative to a class ^V with a develop- 

 ment A', a class ^13", classes di = [p], ® = [a] of functions on '']? = ^].V'i)3", 

 and classes 9}?', 9K" of functions on *ip', ^" respectively. 



/'c'v grj''^'''^'' 5)>"^' <y>Jri'i'!'(r!").£,3isi'iii" m ii:i'm'(i'i").Si2ii'i'i'!" ay x,'KM>i").Si'iiK'(M") 



,Z),Xi' cm//-°) 



(Ml") 



(6) 



(4, 5, 6) are generalizations in the sense of relativity of (1, 2, 3) 

 and the proof follows the same lines. We state (6) as 



Theorem II. // a class 2)2' of functions on ^' has the property Di 

 and the property K\ relative to a development A', and if any second 

 class 9}?" of functions on %" has the property Di ; then the linear extension 

 of the composite class SOJ'W, the extension of 50J'9D?" as to itself, and the 

 *-extension of Wl"^l" have the properties Di, K'i^DVCW). 



The following propositions are generalizations in the sense of rela- 

 tivity of propositions (52, 62, 7i.) of § 23a. 



(9K'a)j") /''=""'*■"'"'. 



Theorem III. If a class 93?' of functions on '^' has the property D 

 and the property K'^ relative to a developynent A' and if any second class 

 9QJ" of functions on "ij?" has the property D, then the classes (Wy)V')L, 

 {Wm").,n", (m'W)^ each possess the properties D, K'M'm"). 



30. Propositions. — Relative to a class %V with development A' 

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



(1) TV 9 {m"'-'^ . 3 . m'm")^'''- "'''■"">) : Z3 : arj'^-'" . ^ . W'--'-, 



viz., if 50r is such that for every W'''^ (it is true that) {Wi'm")^ 

 has the property K'^dil' C^Dl") then if W has the property K[^ it alsa 

 has the property i^Ja*. 



(2) Wi' 9 {m"'-'"' . 3 . (9JJ'9)J")/''™'"''"') : 3 : ^J?'''^'" . zd . W""''. 



