4 pitcher: interrelations of 



properties : 



(6) (++ ••• ++); (++ ••• + -); ••• (-- +); 



( ), 



formed from the properties (4) and (5), the part i of each composite 

 property being + or — according as the property P, is taken positively 

 or negatively. The complete existential theory* of the n properties 

 (4) of systems S consists of a body of 2" propositions, one relative to 

 each of the composite properties (6). Each proposition states that 

 for the required type of systems 2, the composite property in question 

 is existent or non-existent. If the proposition is one of existence it is 

 proved by the exhibition of a suitable system ^. If the proposition is 

 one of non-existence it is proved by a general argumentation showing 

 a relation amongst the properties (4). 



The n properties (4) are said to be cornpletely independent {and 

 mutually consistent) in case the 2" propositions of the complete theory 

 are propositions of existence. The n properties are independent in 

 the usual sense in case there exist n systems S each failing to have 

 precise'y one of the n properties so that no one of the n properties 

 is implied by the remaining n — I properties. 



We show that of the 2^ = 256 composite properties arising from 

 the properties (2), 145 are existent and 108 non-existent. We are 

 unable to decide whether the three composite properties: 



(7) LC-DAAK,K-K,,^; L-C-DAAK,K-K,,^; LC-D-AAK.KfK,,, 



;*» 



are existent or non-existent, but we do show that for certain very 

 extensive regions of the general field they are non-existent. 



Each of the 145 independence examples consists of a particular 

 system (?f; ^; A; Tl). An exhaustive study is made of such systems 

 in the various cases where ^ is 'iJ.V; ^^V'^j sjj'Jjj <|pii» ^nd a complete theory 

 is given for each of these cases. The known 145 existent composite 

 properties are all existent for systems (3(; '$; A; 3!}J) where ^ is de- 

 numerable. 



For the convenience of the reader we give, in the introductory 

 sections, an exposition of tho.-e fundamental notions of the Intro- 

 duction to General Analysis which are essential to the comprehension 

 of this paper, together with those propositions and theorems frequently 



* I. G. A., § 47. 



