PROPERTIES OF CLASSES OF FUNCTIONS 47 



(4) A . a}^^^''"'^^ Z) . TO^"-. 



(5) A-''' . ^^l^-^^''' . 3 . 9JJ^'-. 



(6) A"*' . au" . D . gw. 



(7) A . w-'^^ . 3 . m""'. 



(8.) A . W'"'' . z) . 9[rj^"" (1 = 1, 2, 12). 



43. Propositions relative to ^"' or ^X^"= or *i)?'. 



(1)* A . W''^''''"- . 3 . 9Jr- " 9JJ^''". 



(2)* A . g[yj-^^'^2i^''» . 3 . s)jf^>. - g[)j^-=-. 



44. Propositions relative to ^"". 



(i)t gw^.D.an"^^. 



(2) J A'^^^ 90?^ 3 . 2«^ = 9[RJ\ 



(3)§ A . 9[)J^=^''- . 3 . au*-'-. 



(4)§ A . 9«^'^'" . 3 . a)^^''^ 



(5) A' . m'^'"''''- . 3 . 9Dr-^"'". 



To prove (5) consider ^' as separated into three mutually exclusive 

 classes '^l, %, % defined as follows: 



% ^ [all p; 9 m' . 3 . m;,' = 0]; 



iV = [all pi 3 (3 yu' 3 ju;., + . a ;/ + p; 9 m' • => . Mi'.' = i^p,')]> 



f< ^ [all p: 3 (3 m' ? m;./ + . p' + p: . 3 . 3 m' 3 m; + m;,/)]-!! 



* These propositions follow from a consideration of the various classes "SV^ on ^"3 and 

 then a discussion in each case of the nature of the development A required in order that the 

 class TO may have the other properties required in the hypothesis. 



fl. G. A., §48.7. 



t From a consideration of the developmental system we show that n functions 61 . . . S„ 

 belong to 93! z such that 



011 . . . ^In 



: • +0. 



9„i ... a»„ 



§ 5|3"" may be divided relative to the development into three subclasses '$1, 5p>, ^3 such 

 that A^"!'*!) riCSa) r.(«s)_ Consider the further nature of the development of ifi, 5p2, %\3 and of 

 the functions belonging to S)J as required by conditions on 931. 



II We may think of this classification in another way. 2Ji' is a class of functions on i)J' 

 to 21 but ^' may as well be considered as a class of functions on ffl!' to 3t. In the latter sense 

 ^', is the subclass of ^' consisting of all zero functions on M' to 21. ^',. is the class of all 

 repeated non-zero functions belonging to '$' and $', is the class of all unrepeated non-zero 

 functions belonging to ^. 



