OF AUBOGAST’S METHOD OF DERIVATIONS. 
41 
conclusion is that a contraction on the ultimate and penultimate letters (these being 
non-contiguous) is reversible if the ultimate is a simple letter, or if, being a power, it is 
a power of the last letter. 
Next let the contraction affect the ultimate and antepenultimate letters. The two 
letters are here separated by the penultimate letter, and are therefore not contiguous. 
Suppose that the termination hf^g'^k (/and g contiguous), the contracted term termi- 
nates vaf‘~'^gg”'j. and {g^j) the pair arising from the contraction is the last expansible 
pair of the contracted term ; the contraction is therefore reversible. In the example /* 
has been written, but the case /=! is not excluded; moreover the ultimate letter k is 
taken non-contiguous to the penultimate letter g ; but if the ultimate letter had been 
the contiguous letter Ji, the only difference is that the pair would be [g, g), and the con- 
clusion is not altered. But suppose the termmation of the term is e^g^^k (e, g, non-con- 
tiguous), then the contracted term terminates in where the pair arising from 
the contraction is but the last expansible pair is {g,j)', the contraction therefore 
is not reversible. The case is not excluded; nor is it necessary that the ultimate 
letter should be non-contiguous to the penultimate ; if the ultimate letter had been /g 
the pair arising from the contraction would have been (/ g), but the last expansible 
pair (g, g)^ and the contraction is still not reversible. In each of the cases considered 
the ultimate has been a simple letter : if in the first case the ultimate had been k^{]) >1), 
then the contracted pair would terminate and (according as^.;=2 or^<2) the 
last expansible pair would be (/ k) or {k, k), which is not the pair (g,J) produced by the 
contraction ; the contraction is therefore not reversible, unless indeed k is the last letter, 
in which case it continues reversible. In the second case the contraction, which is not 
reversible when the idtimate is the simple letter k, remains not reversible when the 
ultimate is a power of such letter. The conclusion is that a contraction on the ultimate 
and antepenultimate letters is reversible, if the penultimate and antepenultimate letters 
arc contiguous, and the ultimate is a simple letter, or if, being a power, it is a power 
of the last letter. 
. A contraction on the ultimate and pro-antepenultimate letters, or on the ultimate 
letter and any letter preceding the pro-antepenultirnate letter, is never reversible. To 
show this, it will be sufficient to consider the case where the term terminates in efgh^ the 
contracted term terminates tlic pair arising from the contraction being (/, ^}> 
and the last expansible pair being {(J->g)\ and a fortiori, if the letters or any of theni 
occur as powers, or are non-contiguoiis. 
Consider, next, a contraction on the penultimate and antepenultimate letters; this 
assumes that these letters are non-contiguous. Such a contraction may be reversible 
if only the ultimate is the last letter or a power thereof; and the condition is then 
similar to that in the case of the ultimate and penultimate letters ; only as the penulti- 
mate cannot be a pow’er of the last letter, it must be a simple letter. And the condi- 
tions in order that the contraction may be reversible then are that the ultimate is the 
last letter or a power thereof, and the penultimate a simple letter. 
