PROPERTIES OF CLASSES OF FUNCTIONS 51 



in verifying that any subset of four, three or two completely inde- 

 pendent properties is already included in the subsets of six and five 

 completely independent properties. 



47. Special derelopments of particular classes ^. — In order to be 

 able to give the independence examples previously referred to in a 

 concise form we indicate here certain special developments. The 

 symbolism here maj' appear somewhat formidable, but by a little 

 careful attention the reader should find it perfectly clear. The classes 

 of elements ^ considered are %^\ T\ %V^\ ^^"\ i^'", ^^"■^'", ?"^'" 

 defined as follows: 



^i ^ I ^lu ^ i^ 2. ^lVi3 ^ 1, 2, 3. ^;V'^ = 1, 2, .3, 4. 

 r'' ^ 1, 2, 3, 4, . . ., n,... 



^n.in ^ 3n.^i„ ^ [^j (i = 1, 2; j = 1, 2, . . ., n, . . .). 

 ^ii3in ^ 31133111 ^ [,-j] (,• = 1, 2, 3; j = 1, 2, . . ., n, . . .) 



47a. Developments of '^^. — 

 AI.^.A9(to.3. /,„ = !. i^™ = 1).* 



aJ.^.Aj (m .3./,,, = 1 .-aa^"!).! 



Aj . ^. A 9 (m . 3 . /„. = 2 . ^X'"' = ^;5"'2 = i)_ 



476 Developments of "iP"'.— 



A?'= . =. A 9 (to . 3 . ?„ = 2 . ^^-1 = 1 . l^--- = 2). 



Af' . ^. A 9 (m . 3 . /,„ = 1 . ^>i = 1, 2). 



A?= . = . A 9 (m . 3 . /„. = 1 . -3 i^"*'). 



A'/' . =. A 9 (??i . 3 . L = 1 . 'X"" = 1). 



Ai^= . =. A 9 (to . 3 . /,„ = 2 . ^"i = ^l?""^ = 1,2). 



AJ'' . ^. A 9 (m . 3 . In = 1) . (to°" . 3 . ^"" = 1,2). (to"'" . 3 . ^l^"'^ = 1). 



A?' . =. A 9 {nf'" . 3 . Z,„ = 2 . ^;>i = 1 . ^^"2 = 2) . (to"'" . 3 . Z„ 



= 1 . ^P"! = 1, 2). 



A?^ ^. A 9 (w°" . 3 . ?„, = 2 . ^^""1 = ^T?"^ = 1, 2) . («*'"" . 3 . Z„ 



= 1 . <P">i = 1). 



* For notations 5(5"', Z„,, etc., see § 13. 

 t The class ""^v is tiie null class. 



