44 pitcher: interrelations of 



40. Theoreins.— By § 39al4, 15; §23al6; §29.6; § 9a5; §18.2, 3; 

 § 29.3; § 39a5 the following theorem is readily proved. 



Theorem I. From 



^13 . A . g)J°'^^--«3 (5^ defined in § 39.8) 



it follows that 



This theorem is to be compared with I. G. A.,* § 82II where the 

 property Xij* is secured as a result of D, A, K12, the development 

 being as here unconditioned. The properties A, Ki, K2 are common 

 to the hypotheses of the two theorems. The property Di is less 

 restrictive than D, and it is easier to ascertain whether or not it is 

 possessed by a given class of functions. The genus (31; "5?; A; SOJ-^O 

 is very extensive and includes the systems I; II „; III; IIIo; III^; IV 

 with the ordinary developments of the classes '^ concerned. In 

 particular any class of functions containing a single constant function 

 not zero belongs to this genus. Usually one may ascertain by in- 

 spection whether or not a class of functions possesses the property B3. 

 The hypotheses of the theorem given here cover cases not covered 

 by the theorem of I. G. A., § 82II. Consider for example the system 

 (3(; %^;A;m where 



i^ = interval < p ^ 1 ; 



9DJ = all 2] tti p' where k and I may be any positive integers or 0; 



A = any development of ^]3 such that 90? has the properties A, Ki, Kn. 

 W. does not possess the property D. It might be thought that 

 a development of ^^^ could be designated such that 93? would have the 

 properties A, K^, K^, ~K^^^. However, this class of functions does 

 have the property Di and whatever the development of 1\ 9JJ has the 

 property Bz. Therefore if the development of ^ is such that 9}J 

 has the properties A, K^, Kn, then Wl has also the property iv,,^. 



Theorem II. From 



^B . A . g!}j^<^"i-^^i2S3 

 it follows] that 



m"''-'^ . {BM ~ K,m). 



* Cf. § 23al8. 



t By Theorem I and § 361. 



