384 
Proceedings of Royal Society of Edinburgh. [sess. 
the squared difference-product of the afs is equal to 
h. ■ ■ •><.)} 
t - ( a l + l)^ _ (®2 + l) -(«»+!) 
{ ^(h. 5 ^2 > • • • J In) } 
£-(®l+l)/ _ (<* 2 +l) /-(t*n+l) 
1 v 2 * * ' n 
where the notation u^ed is sufficiently explained by saying that 
in accordance with it the coefficient of x r in the expansion of 
Fhc) is denoted by 
{*(*>}*■ 
Baltzer (1857). 
[Theorie und Anwendung der Determinanten, vi + 129 pp., 
Leipzig: French translation by J. Houel, xii + 235 pp., 
Paris, 1861.] 
The section (§ 12) dealing with the “Product aller Differenzen 
von gegebenen Grossen ” belongs to the second part of Baltzer’s 
text-book, that is to say, the part concerning “applications.” It 
occupies eleven pages, those devoted strictly to alternants being 
the first three (pp. 50-53). 
At the outset he establishes the determinant form for 
the difference-product P(cq , a 2 , . . . , a n ) : then he gives two 
determinant-forms for P(cq , a 2 , . . . , a B ) . P^ , /3 2 , . . . , (3 n ) : 
passes thence to the persymmetric determinants in s 0 , , s 2 , . . . : 
and finally gives Cauchy’s evaluation of the double alternant 
| (cq - /I-l ) -1 (a 2 - yS 2 ) _1 . . . (a n - /? n ) _1 j . The applications, which 
come next, concern the solution of Lagrange’s set of linear equations, 
Sylvester’s transformation of a binary quantic into canonical form, 
and the discussion of the equality of two roots of the equation 
a/-t-a„_/' 1 -l- . . . + a 0 =:0, or say/(#) = 0, viewed in con- 
nection with what he calls the “determinant” of the equation, 
although Sylvester’s use of the word “discriminant” is explained 
a page or two later. 
Under this last head an interesting transformation falls to be 
noted. Calling the roots of the said equation cq , a 2 , . . . , a n , 
and taking the determinant which is the square of their difference- 
product, namely, 
