40 
ME. A. CAYLEY ON AN EXTENSION 
or of the form where c, d are any two contiguous letters, not excluding the case 
where the single letter or each or either of the contiguous letters is a first or a last 
letter. 
It is to be observed that the passage up from any term to the term at the head of the 
column (or top term) by means of a succession of last expansions, is a perfectly unique 
one ; but as no selection has been made of a like unique process of contraction, this is 
not the case with respect to the passage down from any term to the term at the foot of 
the column (or bottom term) by a succession of contractions. 
Every term gives by the last expansion a term above it ; it can therefore be obtained 
from such term above it by means of a contraction. But the contraction of the upper 
term is by hypothesis such that, performing upon the contracted term the last expansion, 
we obtain the upper term ; a contraction, which, performed on any term, gives a lower 
term, which by performing upon it the last expansion reproduces the first mentioned 
or upper term, may be called a reversible contraction. And it is clear that if we perform 
on the top term all the reversible contractions, and on each of the resulting terms all 
the reversible contractions, and so on as long as the process is possible, we obtain with- 
out repetition all the terms of the column. The column is in fact similar to a genealo- 
gical tree in the male line, each lower term issuing from a single upper term, while each 
upper term generates a lower term or terms, or does not generate any such term, and 
the top term being the common origin of the entire series. It may be added that when 
the order of the reversible contractions of the same term is duly fixed, the alphabetical 
arrangement of the terms in the column agrees with the order as of primogeniture (an 
elder son and his issue male preceding all the younger sons and their respective issue 
male) in the genealogical tree. 
It only remains then to inquire under what conditions a contraction is reversible. 
Now as regards any term, in order that a contraction performed on it may be reversible, 
it is necessary and sufficient that the pair produced by the contraction should be the 
last expansible pair of the contracted term. There are several cases to be considered. 
First, if the contraction affects the ultimate and penultimate letters of the term: this 
implies that the ultimate and penultimate letters are not contiguous. Let the term 
terminate in the contracted term will terminate in e”'~^fg^ and (/’, y), the pair pro- 
duced by the contraction, is the last expansible pair of the contracted term ; the contrac- 
tion is in this case reversible. If, however, the term terminate in e”if (2>>1), the con- 
tracted term will terminate in and the last expansible pair is not as before 
(y, y), but it is (according asjp = 2 ory>>2) (y, li) or (/i, li)\ unless indeed li is the last 
letter, in which case (/*, g) remains the last expansible pair of the contracted term. In 
the example e” has been written, but the case is not excluded; moreover, the 
penultimate letter e is removed three steps from the ultimate letter /q but the result 
would have been the same if instead of e we had had any preceding letter, or had had 
the letter jT; by hypothesis it is not the letter contiguous to the ultimate letter h. The 
