PROPERTIES OF CLASSES OF FUNCTIONS 45 



Theorem III. Frovi 



it follows* that the composite systems : 



(31; ^s'^;V'; A'"; 9jr$m"); (21; ^'^"; A'"; {m"m")j); 



(2l;r^";A"';(5mW)*), 

 are such that 



(9D'J'9}J")°"^"''^'''^''^""; (9JJ'9)2")z.^'^"''^'"^"^''"' (SJJ'iDJ") i^'^"'*^i!«3^i2-. 



From preceding theorems and propositions the following closure 

 ■ theorem is readily obtained. 



Theorem IV. The respective genera of all systems: 



0(; ^;^; A; Tl^, 

 where 



are closed under the Jive operations A, B, Co, Ci, d: 

 A : linear extension of classes 9Ji ; 

 B : ^-extension of classes 9)? ; 



C: simultaneous composition of classes ^> and developments A and 

 Co : multiplication of classes 9)? ; 

 Ci: linear composition of classes 93?; 

 C2 : ^-composition of classes 9)?, 

 and the combinations of these operations. 



The respective subgenera of all systems obtained by operations A, B, 

 Ci, Co and their combinations are the repective genera of ail systems: 



m;V;A; a)J^"), 

 ivhere 



Po = 53; LD,Kr2B^; LD,AKi,B,{K,,^). 



These genera are closed under the operations A, B, Ci, Co and their 

 combinations; to these genera belong the syste77is I; II „; III; Ilh; Ilh; 

 IV. 



Propositions relative to systems C^; ^; A; Tl) where '^ is ^^'; ^V or 

 W; 1^ or %^"'- or ^"- ^"", §§ 41-44. 



41. Propositions^ relative to ^^.V. — 



(l)t 9)?.z..9.">r 



* By Theorem I and § 39al6. 



t The propositions of §§ 41-44 are proved by a detailed consideration of possible 

 systems satisfying the specified hypotheses. 

 % I. G. A., § 4S.9, 10. 



