18 pitcher: interrelations of 



In the theory it is only the ultimate character of the development that 

 is of importance. 



(5) A^=. = . A 9 (m . 3 . 3 ?«n > m j A'"»0, 



viz., the development A has the property P2 in case for every m there 

 exists an TOq > m such that the stage A"^ has the property P. 

 Thus we have the properties,* 



V (vacuous), Vi (ultimately vacuous), F2; ^ (fine), Fi (ultimately fine), 

 F2; C (complete), Ci (ultimately complete), Co, 



of developments A, derived from the properties (1) of stages A". 

 The developments A', A"", A'" of §15 are fine developments while the 

 development A'^ of the same section is a complete development. 



In various propositions and proofs of the sequel the negative of 

 such properties P, Pi, Pi enter. These are in notation: ~P, "Pi, ~Pi 

 and accordingly: 



(6) A"-^. CO. A? (gr^jA^'O, 



(7) A"-^' . ~ . A 9 (m . Z3 . 3 rHo > m s A™'-^), 



(8) A'-^^ ~ . A 3 (3 Too i7n > mo.^. A^'O- 



The properties ~P, ~Pi, "Po must not be confused with the properties 

 (~P), (~P)i, (~P)2 concerning which, according to the general defini- 

 tions, we have : 



(9) A<'-P'.~. A9 7?j .D. A'""-^, 



(10) A'"'^'> . ~ . A 9 (3 Too 3 TO > 7^0 . 3 . A'''^, 



(11) A'-'''" . ~ . A 9 (m . 3 . 3 Too > m 9 A'""0. 

 16a. Propositions. — 



(1) A^.o.A^ (4) A^3.A'^^. 



(2) A^'.^.A^^ (5) A''.z). A^'"^'. 



(3) A ^= . ~ . A^'^''. (6) A''^.r).A^^ 



17. Developmental systems^ ©(SO'?) relative to a class 9)^ of functions 

 and a development A. — A developmental system J)(2R) : 



(1) ©(5DJ) - ((6-0), 



* The properties fine and complete of developments A are defined by Moore in I. G. A. 

 § 67b. The property vacuous is likewise defined in I. G. A., § 75. 

 tl. G. A., §78. . 



