298 Mr. Jourdain on the Transjinite Cardinal 
In fact, consider a double series (w,,), of which each line 
and each column is of type @,: 
Hw “4A, w4 ++ Uy py | 
Le CER Mima MON cb rressbe G8 5. ao | 
| 
Uo Uno «+ Yow? Yo, or Uy, pr se | 
TT U U u She AOE oid r 3 (2) 
M+1, 2 “w+1,2) -°* “w41,w? “wtl w+) +1, p? | 
| 
Ce it) CA eee Lae w? Un w+1) iat at UB? | 
we shall determine a series (a,) of type , such that there is 
a@ one-one correspondence between this (simple) series and 
the double series (w,, g). 
The series (ay) is formed by diagonal enumeration of (2) ; 
thus to the terms 
Uy 19 Uy 99 Uo 1 Uy 39 Uy o --- 
correspond respectively 
Og lo Cee (One Og 
and the general relation between (a,) and (w,,¢), when 
a, 8, y are finite integers, is * 
y=ath(a+t@B—-l1)(2+B—2). . . . (8) 
When «@ and 8 are such that 
bia<o,.0=f< a2; 
and 
oXa<ow.2, 1<B<e, 
the manifold (ugg) is still of cardinal number No, and can 
consequently be made to have a one-one correspondence with 
the series 
a a 
wr Concer: 22> Cog ero 
* This formula was stated by Cantor, Journ. fiir Math. Bd. lxxxiv. 
(1878), Satz (C’) of §8 ; Math, Ann. Bd. xlvi. p. 494; and a simple proof 
and a generalization for multiple series was given by me (“ On unique, 
- non-repeating, integer-functions,” Mess. of Math. May 1901). 
