214 Proceedings of Royal Society of Edinburgh. [sess. 
from every column of A , and will therefore be a term of A if only 
we can show that the number of inversions of order in 
2,1, 4,3, 6,5, . . . , w, ra — 1 
is \n , a fact which is self-evident. 
Baltzer’s proof that the rational integral function H, which is 
the square root of A , changes signs when two suffixes, r and s, are 
interchanged is a simplification of Brioschi’s, the operation and 
even the notion of differentiation being dispensed with. The 
function resulting from the change being H' he concludes like 
Brioschi that 
H 2 = H' 2 ; 
also the aggregate of the terms in H which contain a rs being a rs B, 
say, he infers as Brioschi does that B cannot be affected by the 
change, and that therefore a rs B will be altered into a sr B or - a rs B. 
Here, however, he brings the demonstration quickly to a satis- 
factory end by saying that since some of the terms of H' are thus 
seen to differ in sign only from the corresponding terms of H, the 
equation H 2 = H' 2 shows all of them must so differ ; and this is 
what was to be proved. 
Jacobi’s notation for the function H is then introduced, the 
formal intimation being that (1, 2, 3, ... , n) is used to denote 
the aggregate whose first term is a 12 a M . . . a n _ 1>n and whose 
square is A . The other value of J A is thus of course represent- 
able by (2, 1, 3, ... , n), (2, 3, . . . n, 1), or . . . As this implies also that 
JA' rr = ± (2, 3, . . . , r - 1, r + 1, . . . , n) 
we have now the means, so far as symbolism is concerned, of 
removing the ambiguity from the various terms of the identity 
\f A ~ : ®12 \A^- 22 4 33 . • . + ^in\/A nn > 
As for the knowledge necessary to use the symbolism aright, 
Baltzer’s dictum is that the sign taken to precede (2,3, . . . ,r - 1, 
r + 1, . . . ,n) in substituting for JA' rr must be such that the equation 
JA rr . f A ss — A' 
will be satisfied ; and this he proves will take place when the 
