PROPERTIES OF CLASSES OF FUNCTIONS 67 



given in § 48 of this paper show that the properties D, A as properties 

 of classes g)J'*-'i*^='^i=' are completely independent. Many of the cases 

 where the property 'DK^„^ is present are covered by the theorem of 

 § 401 of this paper. 



As before remarked we have neither examples illustrating the 

 composite properties, 



LC-DAAK^K.-K,,^; L-C-DAAK^K^-R,,^; LCrD-AAKiKo-K,,^, 



nor propositions showing that they are non-existent. However, by 

 theorems §321; §351; §381; §401, these composite properties are 

 non-existent for certain important and very extensive parts of the 

 general field. Thus we have the following theorem: 

 Theorem. The three composite -properties: 



LC-DAAK,K.-K,„^\ L-C-DAAK,K.-K^,^; LC-D-AAKiKo-R^,^, 



are each non-existent for all systems: 



(31; ^;v A^'- m, (?l; V; A^'; m), (3I; ^i>; A; 9jr*). 



