42 
ME. A. CAYLEY ON AN EXTENSION 
The next case is that of a contraction on the penultimate and pro-antepenultimate 
letters. Such contraction may be reversible if the ultimate is the last letter or a power 
thereof ; and the condition is then similar to that for the case of the ultimate and ante- 
penultimate letters ; only as the penultimate cannot be a power of the last letter, it must 
be a simple letter. The conditions, in order that the contraction may be reversible, are 
that the ultimate may be the last letter or a power thereof, the penultimate a simple 
letter, and the antepenultimate and pro-antepenultimate letters contiguous. 
A contraction on the penultimate letter and on any letter preceding the pro-ante- 
])enultimate letter, or upon any two letters preceding the penultimate letter, is never 
reversible. If the ultimate, penultimate, antepenultimate and pro-antepenultimate 
letters are denoted by U, P, A, P' respectively, then by what precedes, the following 
contractions, viz. UP, UA, PA, PP', may be re^•ersible, and they will be so under the 
conditions shown in the annexed Table. It is to be noticed that the conditions for UA, 
PA are mutually exclusive, and consequently that the number of reversible contractions 
to be performed upon any term is at most 3. The Table is 
U 
P 
The contractions are to be applied in the order UP, UA, PA, PP', but all the con- 
tracted terms originating in a prior contraction of a given term are to be exhausted 
before proceeding to a posterior contraction of the same term. As an example of the 
process, the set being («, c, e), I will take the column 
ahe^ 
acde^ 
ad^e 
Ifde^ 
bc^e^ 
bcd'^e 
&de 
C'd^ 
The top term abd' is given. The contractions applicable to it are UP, UA. And UP 
gives acdd. The only contraction applicable to this is UA, giving ad?^. And there is 
PA P' 
U, P, non-coutiguous letters. 
The ultimate a simple letter, or 
a power of the last letter. 
P, A, contiguous letters. 
The ultimate a simple letter, or 
a power of the last letter. 
P, A, non -contiguous letters. 
Penultimate a simple letter. 
Ultimate the last letter, or a 
power thereof. 
A, P', contiguous letters. 
Penultimate a simple letter. 
Ultimate the last letter, or a 
power thereof. 
