108 Proceedings of Royal Society of Edinburgh. [sess. 
Lastly, attention is drawn to the case where a = 0, h= 1, s=l, 
and to a case where the order of the alternants is infinite, viz., to 
the fraction 
The fifth and last chapter (pp. 395-398) concerns the simplest 
form of alternant above met with, viz., that in which the indices 
proceed throughout by a common difference, the main proposition 
being regarding the resolvability of the alternant into binomial 
factors. The property with which Cauchy and almost all later 
writers start is thus that with which Schweins ends. The mode 
of proof is interesting from its farfetchedness and ingenuity, but 
need not be given in full generality or in the original notation : 
the case of \a% l c 2 d^\ will suffice. 
The first step, then, is to select a row, say the last, and express 
the alternant in terms of the elements of this row and their 
complementary minors. In this way we obtain 
\a°b l tfd*\ = d*\aW\ - d 2 |aW| -f d\a Q b 2 c s \ - |a W| . 
Now each of the alternants on the right is expressible as a multiple 
of | oWc 2 ! by means of the theorem above given regarding alter- 
nants with one break in the continuity of the equidifferent pro- 
gression of their indices. Using this we obtain 
1) a a+h a+2h 
|a 0 JW| = {d 3 -d\a,b, e) l + d(a,b, c) 2 - (a, &,c) 3 } . |aW|, 
= {d 3 — cP(a + b + c) + d(ah + ac + be) - abc} . | aWc 2 ! , 
= (d- a)(d - b)(d - c) . |a°i 1 c 2 | , 
