Vol. 8, 1922 MATHEMATICS: C. N. MOORE 291 
and with regard to the functions <^ we postulate 
(F) all functions ^ are of the class g. 
We also define the notation 
(1) (D^) (a) = a(a) 
with regard to every B, a on @ to to mean that 
We further postulate as to the class ® 
(i?) a' ■Z>-n {<r\)-r = <r' (<r'), 
for the class ^ 
(C) 77- C7'-D. Hlim ,7((r)=^(<70 (?), 
and for the class g 
(A) a z?p - [(Ov) (<r) ki^- 
For the operation / we postulate 
(Id) v=>-{DUv)) W = vM W. 
We wish finally to postulate certain properties of the special function v?o. 
We need first to introduce a further definition. We shall mean by the 
notation 
lim 17(0') = 00 
cr 
that for every positive e there exists a o-^ (dependent on e) such that for 
every (t>g^, 77(0-) >e, or in symbols 
This definition also is based on E. H. Moore's definition of a limit referred 
to above. We also introduce for the sake of brevity the following notations 
(2) ipQn{(^) = (Po{(t) • (po{ai) ■(pr{a2) • • • • (poicrn-i) 0'^>1), <Poi{o') = <PqM, 
(3) ^PnM = (n>l). 
<PO,n-l {(^1) 
We now postulate for (po that 
(I) (Po is a real and positive function of cr, 
