48 pitcher: interrelations of 



Further separate '^l into classes "iV,,., (i = 1 • • • q), such that: 



p'i*°^''''. 7^/°V. Si ^ Sj .:d .3 /i' 3 Mi'..' 4= Mm" 



™y, ,a:i'A-2'A' . , Ci( iv'+ li.')VnB/J^'i( iV) 



Also since W'^'' the development must be such that any two ele- 

 ments p'rp', or p'rp'i or p',p', or 2^1,Ps, are ultimately separated by the 

 development. 



W''-^'. %r ^ • 3 mo i (m ^ Wo .3.3 1'" 3 %, = ^V"), 

 Having thus characterized the development it is not difficult by a 

 consideration of the developmental system and a determinantal 

 consideration as in (2) to show that 



5m;, = [all m' 3m' = constant {%) (t) . m' = i%K)]. 



= [all ^^"'^'"'""j ^^„ = constant {%K,) (ip") . <py, = (^V,) (p")]. 



gi ^ [all ^*-vs''c^'"'.s„K",;/;] = [all <p«"'""*'', ^,,„ = {%) ip")]. 



ga = [all ^•K/="'(2"").so™"(p';] = [all ^iio-j^i'w , ^^„ = constant (%;) {ip")]. 



gii, = [aU ^A--'»'c>"")-«o-^'"»'.'] = n (gig.) = (aR'9«")*. 



r/ie complete existential theory of the eight properties: 



L; C; D; A; A; K,; A%; A',,^. §§45-48. 



45. The complete existential theory of the properties L; C; D; A. — 

 Moore shows* that the properties L; C; D; A are completely inde- 

 pendent.! Of the 2^ = 16 composite properties formed from L; C; 

 D; A and their negatives it is shown that only five, viz., 



(+ + + +); (- • + ■),t 

 are possessed by classes of functions on 1^ singular and only ten, viz., 



(+ + + •); (- • • •) 



are possessed by classes of functions on ^1.^ finite. 



* I. G. A., § 48. 



t See I. G. A., § 47 and preface to this paper for definition of complete independence. 

 t The dot • indicates that the corresponding composite property may be taken either 

 positively or negatively. 



