OF AEBOOAST'S METHOD OP DEEIVATIONS. 
39 
Some definitions and explanations are required. I apeak of the first and last letters 
of the set, simply as the first and the last letter : there is frequent occasion to speak of the 
last letter ; and to avoid confusion, the last letter of a term will be spoken of as the 
idtimate letter; it is necessary also to consider the penultimate, antepenultimate, and 
pro-antepenultimate letters of the term. It will be convenient to distinguish between 
the ultimate letter and the ultimate, which may be either the ultimate letter or a power 
of such letter ; and similarly for the penultimate, &c. Thus in the term hcd?, the ulti- 
mate letter is d and the ultimate is d^, the penultimate and the penultimate letter are 
each of them c ; of course the ultimate, penultimate, «&c. letters are always distinct from 
each other. We have also to consider the pairs of letters contained in a term; cd^e con- 
tains the pairs (c, d), (c, e), {d, d), (d, e), and so in other cases ; the letters of a pair are 
taken in the natural order. A pair not containing either the first letter or the last 
letter is expansible ; thus the set being as before {a, h, c, d, e), the pairs [h, c), [d, d) are 
expansible : they are expanded by retreating the prior and advancing the posterior letter 
each one step ; thus the just mentioned pairs {b, c), (d, d), are expanded into [a, d) and 
{c, e) respectively. 
A pair composed of tAvo distinct non-contiguous letters is contractible ; it is contracted 
by adA'ancing the prior and retreating the posterior letter each one step : thus [a, d), 
[c, e) are contractible pairs, and they gAe by contraction the pairs (b, c), (d, d) respect- 
wely : the processes of expansion and contraction are obviously converse to each other. 
The expression the last expansible pair of a term hardly requires explanation; {d, d) 
is the last expansible pair of the term bcd^, or of the term (c, d) the last expansible 
pair of the term C'de (the set being ahvays {a, b, c, d, e)), and so in all other cases. 
The expression the last expansion, in regard to any term, means the expansion of the 
last expansible pair of such term. 
The expansion or contraction of any pair of a term leaves unaltered as well the weight 
a's the degree, the resulting term belongs therefore to the same column as the original 
one. But the effect of an expansion is to diminish, and that of a contraction to 
increase the alphabetical rank, or rank in the column, the ranks being reckoned as the 
first or lowest, second, third, and so on, up to the last or highest rank. In particular, 
by performing upon any term the last expansion, we diminish the rank in the column, 
and by a succession of last expansions we bring the term up to the head of the column. 
Such term is not susceptible of any further expansion ; it may therefore contain the 
first and last letters or either of them, and it may also contain a single intermediate 
letter in the first power only; thus the first and last letters being a, e, the head term of 
a column is of the form or ai'ce'‘, c being some intermediate letter, and the powers 
e^ being both or either of them omittable. In like manner, by a succession of 
contractions of any term w'e obtain the term at the foot of the column; such term 
is not susceptible of any further contraction, and it must therefore be composed 
either of a single letter or of two contiguous letters, that is it must be of the form c*”, 
G 2 
