302 Transjinite Cardinal Numbers of Number- Classes. 
we can say that 
at oF 
But, on the other hand, rhe 
qa< (24)8 — 24 
since * 
24> and a’=a. 
Hence 
qt 2". 
But further, if 
b< 24, 
we have also ij 
pW ee YAO 
where 0 is any cardinal number such that 
1<O0<D. 
In particular, if A=N,, we have 
x®y ao¥y, 
y+1 
If a, BD, and D are any transfinite cardinal numbers, then 
(b +0)*=(b.d)*=b*.D3; 
which reduces to b#, if we suppose B20. This last result 
has been proved by Whitehead without utilizing the properties 
that 
b+0=b.d=D. 
Whitehead + defined a class of cardinal numbers, which he 
called ‘‘ exponential numbers,” as follows :—An exponential 
number isa cardinal number @ such that there is a transfinite 
cardinal number 2B such that there is a cardinal number 
Q > 1 such that 
a=. 
Further, Whitehead called those exponential numbers in 
which also 
b = &> S b 
* Detailed proofs of the pa es - if a, b, 0 are any three cardinal 
numbers, and 6<D, then . 
ab<a> and be < 3%, 
were given by Russell (see Whitehead, Amer. Journ. of Math. xxiv. 
p- 382, propositions 4.4 and 4.41, and p. 368). 
+ Amer. Journ. of Math. vol. xxiv. (1902) p. 391, proposition 20. 1. 
The propositions 19.7 and 21.1 to 21.4 can be at once verified by our 
general theorem, but it is interesting to observe that Whitehead proves 
them without using the properties resulting from the fact (which was 
proved in my first paper) that every cardinal number is an Aleph. 
