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Proceedings of Boyal Society of Edinburgh . [sess. 
in all probability the same as before, viz., Laplace’s expansion- 
theorem. The most general form of the latter theorem, it will be 
remembered, gives an expansion in terms of products of more than 
two minors. Jacobi was familiar with this, for in his famous funda- 
mental memoir regarding general determinants a whole page (pp. 298, 
299) is devoted to an illustration of it. Now, if we take the case 
where the number of minors is three, and apply it to the determi- 
nant which is the equivalent of the difference-product, we obtain a 
result which is transformable without difficulty into 
n(« ( 
o, ”!» 
= 2 ± 
/ 
On) 
( x II(a 0 , aj, . . 
. . a k y+ l (a k+ ia k+ z . . . af) k+1 \ . 
ai)U(ai + ia i+ 2 • . a k )U(a k+ ia k+ 2 . . . On) )’ 
and this is the theorem “ of still greater generality ” above referred 
to. 
Jacobi then proceeds to the consideration of alternating functions 
in general. 
The definition which he gives, and to which he attaches 
Cauchy’s name, is somewhat different from Cauchy’s, being to the 
effect that an alternating function is one which, by permutation of 
its variables, is either not changed at all, or is changed only in 
sign. 
In the matter of notation he also introduces a variation, but 
this time with more success. It will be remembered that, when 
Cauchy denoted a determinant by prefixing S ± to the typical 
term, he was simply following his practice in regard to alternating 
functions in general, which he denoted by 
S ± <f>(a,b,c, . . ., 1), 
the rule for determining the sign of any term of the aggregate 
being left unexpressed. Instead of this, Jacobi uses 
<b(a,b,c, . . ,,l) \ 
P P 
where P stands for the product of the differences of a, b, c, . . ., l- r 
and as the P which is inside the brackets is subject to permutation 
of its variables, and therefore automatically, as it were, changes 
sign with every interchange of a pair of variables, while the P 
which is outside the brackets remains unaltered, it is clear that 
