PROPERTIES OF CLASSES OF FUNCTIONS 53 



47/. Developments of 'ip^^iii — Corresponding to the composition of 

 the classes ^"-, ^13"' we introduce the composite index ij for I so that 

 stage m has l,„ = /„j"„ subclasses 'iJ?'"'-'', i = 1, 2, . . . i„r, j = 1,2, . . . j,„. 



A}'''" . = . A 9 (m : 3 : {^ = 2 . j„, = m . l^""' = ij (ij)). 



A?^'" . =. A 9 (m . z) . u = 1 . j„, = 1 . a^""' = 11, 21). 



Ai^^"' . =. A 9 (m . D . u = 1 . j„. = 1 . ^;V"" = 11). 



A?'"^ . ^. A , (/« . D . {„ = 1 . 7;, = m . f-^' = Ij, 2j (j)). 



A?^™ . ^. A , (m : 3 : (i„ = 2 . J,„ = w) . ( j ^ 2 . J . 3 . ^lV«o- = y) 



. if"'' = ^;V"2i = 11, 21)). 

 AJ'='" . =. A 9 (w : 3 : (?",„ = 2 . ./„, = ni) .{j<m.i.^. l^-'^' = ij) 



_ (^^mim = i„j _ s^,m:m = 2m, 2m + 1, . . ., 2m + n, . . .)).* 

 AP='" . =. A 9 (?», : 3 : {i„, = 2 . j„ = m) . (i . j 4= 1, m . 3 . f^"'^' = y) 



. (sp"-!! = s)}m2i = 21 . ^s"'"" = 1 m . ^;s'"-"' = 2m, 2 m. + 1, . . .)). 

 AJ'="' . =. A 3 (m"" . 3 . A" = Aj'="'-) . (m""" . 3 . A"" = AP''"-). 

 A^'^'" . =. A 9 (m : 3 : (,/„, = 2 . j^ = m) . (j + 1, »« . t . 3 . ^;^'"'^" = y) 



O^mll _ \nm21 = 11 21 . >'Ts'"l'" = Im . l?"-'" 



= 2m, 2m + l,...,2m + n )). 



A5f " . =. A 9 (rw"'" . 3 . A" = a;'='"'") . (m"'" . 3 . A" = Aj'="'-)- 



47g. Developments of '^"^"\ — 



AP»'" . =. A 9 (m . 3 . i„ = 3 . j„ = m, . ^li^""' = ij (ij)). 



AP="' . =. A 9 (m .3.4 = 1. j„. = m . ^^'^ = Ij, 2j, 3j (j)). 



Al''"i . =. A 9 (m . 3 . i. = 1 . j,„ = 1 . ^1?™" = 11). 



A}'="' . =. A 9 (m : 3 : (u = 3 . j,„ = m) . ( j > 1 . i . 3 . ^;s""> = ij) 



.(1.3. i^-"'! = 11, 21, 31)). 

 AV^'" . ^. A 9 (m . 3 . i„, = 1 . ;,„ = 1 . ^^«ii = 11, 21, 31). 

 Af'^" . =. A 9 (wr" . 3 . A™ = AJ'^''"'" ) . (m°'=° . 3 . A"" = aP»'""'). 



48. Examples. — In ,the following sixteen sections, 48a-48/j, we 

 give the 145 independence examples previously mentioned. We clas- 

 sify these examples into sixteen sets. Each set is characterized by 

 one of the sixteen composite properties formed from the properties 

 L; C; D; A and their negatives and with this restriction contains the 

 examples illustrating all known existent composite properties formed 

 from the properties A; K^; K„; K^,^ and their negatives. § 48a has 

 the caption: 



* 2 TO, 2 m + 1, etc., are bipartite indices of which the iirst part is 2 and the second 

 part is m, m + 1, etc. 



