38 pitcher: interrelations of 



The subge7)us of all systems obtained by operations A, B, Ci, d and 

 their combinations is the genus: 



(2t; %^; A^'; Wn, 

 where 



P„ = LDAK,{K,)KaKu*). 



This genus is closed under the operations A, B, C\, C^ and their combina- 

 tions. To this genus belong the sy sterns: 



I; n„; III; III,; IIIo, 



in each case the development of the system being the ordinary development 

 of the corresponding class %^. 



Here the properties in parenthesis are impHed by the other con- 

 stituent properties of the respective composite properties in view of 

 the hj^pothesis A'"' and propositions § 27.10; § 20a2. 



36. The theorem of this section is to be compared with Theorem 

 III, § 82, of the Introduction to General Analysis.* It follows readily 

 from § 23al5, I12. 



Theorem I. Fro77i 



"iB; A; yji^^^'^^^i^ 

 it follows that 



m^ . y)} = [all c^^'-'"] . {BM ~ K,.m). 



37. Notation and definition. — A function ;u on '"]? with development 

 A is said to be ultimately zero as to the development, in notation: 



,,ult 0(A) 



in case there exists a stage m of the development such that if p is 

 undeveloped for any stage m^ ^ m then /^p = 0. That is, 



m""°<^' . =. M s (3 m 9 A'p,„ . =5 . ;up = 0). 



It is to be noticed that in case of 'ij.^"' with ordinary' development this 

 definition of ultimately zero coincides with the ordinarj' notion of 

 ultimately zero. 



38. Propositions. — 



(1) f .A. ip''"'"^\m.zi. ^p"'"". 



(2) '^.A. 9jr''- . z) . [all m"" «^']-bo-^'-. 



(3) f .A. 2)J^^^'- . z) . [all m"" "'^Y'"'. 

 * § 23al8. 



