PROPERTIES OF CLASSES OF FUNCTIONS 



61 



(2) - + + -;- 



(3) - + + +;- 



(4) +- + -;- 



(5) -- + -;- 



(6) + + ; + 



(7) - + --:- 



(8) + ; - 



(9) ; - 



and functions juo and 6 such that 



Moi; = - M02y = 1/j- ij) ■ Gij = 1// (y). 



Proof of 9)J^'^'. — This follows readily from the proposition* 



<^,n.ni. ^„=iii. gjj^ ^ [alU«»» 9 Mu = M21]. 

 .^,11,111. ^n.iii. gjj^ ^ [allM^»^"9^u = /X21]. 



sj^iwii. ^ii=iii. g;,., ^ [all m''""' 9 Mil = 0]. 



a^"='"; AP'"^; W2 ^ [all m^""' 3 /xn = 0]. 

 v;Vi="i; AF='"; 9JJ2 ^ [all /x''"'-" ? Mn = 0]. 



48n. L-C-DA, 9 examples (§23alli2, 17), f 9 on ^''" (§44.1), 

 9 on IV".— 



(l)t + + + +;-- -F^A'^"; 



nrj' = [all m' 9 (3 A. 9 j ^ j;- . ^ . (iu'.,- = o . m^, -i^vj)] 



= [all M 'Ml ■ M2 J-§ 



Proof of W''"'. — The classesjl gi, go, S12 here are easilj' com- 

 puted. The classes W of the examples of § 48/;l and § 48Z1 each 

 possess the property ivj,*. In each of these cases the classes ^12 and 

 therefore the classes (9Jf'i[>t")* are easily computed. From these 



* I. G. A., § 23c8. 



t The reference cited permits ten examples in this set, but we have neither an example 

 for the composite property (L~C~D A A KiK^'Kiu) nor a theorem precluding the ex-istence 

 of such an example. 



X An interesting example (see I. G. A., §4Sd) for tliis case is: 



%^]ij] (i = 1, 2, . . ., n, . . .;i = 1,2, ...,n, .. .); 



A 3 TO . 3 . (u = y„ = m) .(iKm.j.Zi. '^"•'' = ij) . (j . 3 . $"""' 



= mj, m + Ij, . . ., ?n + nj, . . .); 

 in = [all M ' (3J„ 3 J SiM ■ => • M.i = 0(0) . (j : 3 : 3 1, » i S ii . 3 . ;i<i,- = Mv/)]- 



This example is obtained from the examples of § iSkl, § 48/1 by the regular composition 

 and linear extension of the classes 2)c and the regular composition of classes ^^5 and develop- 

 ments A. The example given above is an example of composition of a different kind. 



§ § iSkl, § 4sa. 



II Of. § 48. 



