220 Miss Wrinch, On the exponentiation of well-ordered series 



all these subsets of \, i.e. the class of relations which belong to all 

 the sets 



\, 2 P \, 1 P 1 P \ . . . ', 



and get a subset of X 



p'iT'pW^ 

 which, again, consists of members of \ which come earlier in the 

 P'^-order than members of A. not belonging to it. Repeating the 

 original procedure we get 



Tpy(%h'\ Tp'TpYiT'p)^'\, ..., 



and so obtain a series of sub-classes of A, ordered by the serial 

 relation 



A {Tp, \), 



where A is the relation between /j, and v when v is contained in /i 

 but not identical with it. And this is a well-ordered relation : 

 CQjisequently it will have an end, viz. 



p%Tp^Ay\. 



If this is not null, it consists of a single member, which will be 

 the minimum of X in P^. But if it is null we will put 



PQ'X = s'N {a^ . ^,e (Tp^Ayx . iV= (i^/.) r eQm VI- 



Then PQ'X is a relation covering a certain section of the Q-terms 

 with P-terms : PQ'\ agrees in the way it covers the Q-spaces with 

 each member yu, of ^ {Tp, \) as far as QniV- PQ''^ will therefore 

 cover Q-spaces up to z, if there is a yu- which is a member of the 

 field of 



A (Tp, \) 



such that z precedes QmV in the Q-order. If no member of the 

 field of J. (Tp, X) agrees in the covering of Q-spaces beyond a cer- 

 tain member z of the field of Q, PQ'X covers no spaces beyond z 

 with P-terms and for this reason is not a member of the field of P'^. 

 If P is a P'3-term which agrees with PQ'X in the covering of 

 Q-spaces as far as it goes, R precedes all the members of X, in the 

 P^-order ; further, any member of the field of P^, following R and 

 all relations agreeing with PQ'X as far as it goes, follows at least 

 one member of X. Hence, if there were a maximum in the P'^- 

 order in the class p of members of the field of P*? which agree with 

 PQ'X as far as it goes, this relation would precede all X's and any 

 relation following it would follow at least one member of X. If the 

 class consists of one term R, it will have a maximum, namely R 

 itself: R will then be equal to PQ'X and PQ'X will, therefore, be 

 the lower limit of X. But p is a unit class only when PQ'X covers 



