PROPERTIES OF CLASSES OF FUNCTIONS 23 



of the respective classes: 



By the composition* for every stage Wl of the II classes of stage m 

 of A' with the l',',, classes of stage m of A" we secure the 



classes of stage ??i of a development A, in notation : 



A'A" or A' ", 

 of the class: 



the composite develo-pment A'A" = A' " of the developments A', A". 

 For notation it is convenient to use for index i™ or I of stage m the 

 double index r"T"" or I'l". Then 



A'A" ^ A' " = ((r")) = ((F'"")) 

 where 



In - I'JL (m); ^"""" = ^''"" l^"""" {ml'l"). 



Similarly, from two representative systems: 



9t' ^ ((/""')); 9?" = ((r"""")), 



of A', A" respectively we obtain a composite system: 



9t'9i" = 9{"' = ((O) = ((r'"''"')), 

 where 



r = ? J- {mit ), 



in fact, a representative system of the composite development 

 A'A" = A' ", the composite representative system di'dl" = di' " of the 

 representative systems 9?', 9f". 



Similarly from two developmental systems: 



®' - 2)'(2rJ') -^((5'"'")); S" ^ S)"(9)r') = ((5"""")), 



relative to (A', 9)J'), (A", W) respectively we obtain by multiplication 

 a composite systerti: 



3)'®" = ®"' = ((5"")) ^ ((5""''")), 

 where 



8-'''" ^ s'""V'""" (mrr'). 



In case 9D^' has the properties DiA'A'i, and W the properties DiA"K'2 



* The composite of two classes is non-existent if either component is non-existent. 



