PROPERTIES OF CLASSES OF FUNCTIONS 15 



equally well to classes of functions on ^'^" to 31 and in particular 

 to the composite class WW on ^'f" to 21. 

 12a. Propositions* — 



m'wy.m'W)^^ = m'Ti")^ = m'^^m'^)^ 

 m'w)^ = m'M'DL = m'M')^ = (wmi),. 



13. Developments A of classes^ ''^>. — A development A of a class ^ 



of elements is a sequence {A"*! of systems A"" of subclasses of *;}?, each 

 system A"" consisting of a finite number, say Z„„ of subclasses of '^. 

 The system A"" is stage m of the development A. The sequence \lm] 

 of positive integers is the sequence of the development A. 



There is no question of order, but there may be repetitions amongst 

 the Im subclasses of stage m, and it is to provide for this that we say 

 system A"" instead of class A". 



There is introduced an index I"': 



^ = 1. 2, •••, Z 



mj 



of stage m, but the superscript m of the index Z"* is omitted, when in 

 conjunction with the index m of the stage, it occurs in the bi-partite 

 index ml. The Im subclasses of stage ?« are denoted respectively by 

 the notations: 



Then stage m of the development A is the system: 



A" = (i^"')) 

 and the development A is the system : 



A ^ ((^W)), 



of subclasses ^"'^ of the class "ip. 

 A representative system: 



m^ = ((r™')), 



of a development A is a system of elements r""' of *il3, the element r"' 



* Cf. I. G. A., § 54a. D and Di are examples of unipartite properties invariant under 

 multiplication of classes. 

 tl. G. A., §75. 



