54 pitcher: interrelations of 



LCD A, 9 examples (§23all,2, 17), 4 on 'iV (§ 41.6, 7, 12), 9 on ^s"--. 

 That is, in this set are given nine existent composite properties 

 in each of which the properties L; C; D; A occur positively. The 

 propo.sitions §23olli2, 17 show that there are no other existent 

 composite properties similarly characterized as to L; C; D; A. Four 

 of the examples of the set are on ^^ and the propositions § 39.6, 7, 12 

 show that the other composite properties of the set are non-existent 

 for P^. All of the composite properties of the set, nine in number, 

 are existent for '^^"^ Only five of the examples are given on "ip''^ 

 but those composite properties existent for ^^ are also existent for 

 ^ii2^ '"1?";., s;V", etc. In fact the character of a class W as to the 

 properties 



L; C;D;A;A;Ki;K,;K,,^, 



is invariant* under the enlargement of '^ to a class "ip' bj' the adjunc- 

 tion of a class ^" having no elements in common with ^, the de- 

 velopment of ^1^' being identical with the development of ''1? and the 

 functions ii of M being extended to functions ;u' of TV vanishing on ^". 

 We give each example in the following order: 



(a) character as to A K^ K^ if ,2*; -^'i* -^'2* in^ the order indicated; 



(b) The class "ip of the example ; 



(c) the development A of the class '^ ; 



(d) The class 2Jt designated as a class or by the enumeration of 

 the functions belonging to the class; 



(e) Proofs are given where it seems desirable. 



In all the examples in which the property A occurs positively, the 

 developmental system satisfies the conditions la, I'b of § 17 and is, 

 therefore, independent of the representative system. The reader will 

 have no difficulty in specifjdng a developmental system 3) (9}?) such 

 that for every m and p 



X5;" = l; J:A57 = 1; S;/: = m. 



'J ii 



In case the development A is vacuous the reader should recall that 

 every class W has the property A. We do not in general give the 

 developmental systems for the examples, but we do indicate, in foot- 

 notes, developmental systems for the examples § 48a8; § 48firl; § 486^10 

 which seem to be illustrative. 



In case the property A occurs negatively, to validate the example 



* The same phenomenon as to L; C; D; A is noted in I. G. A., §48. 



