1887.] Dr T. Muir on the Theory of Determinants. 485 
pair, et le signe negatif lorsque ce nombre cle variations est 
impair ; enfin, prenez la somme de tous ces produits separes. — 
Yous anrez ainsi, par exemple, 
^[A^XJ - A“Xj , 
«?[>*! . A 6 X 2 ] = A a X r A & X 2 - A 6 X x . A a X 2 , 
The new name, combinatory sum , and the new notation, did not 
originate in ignorance of the work of previous investigators, for 
memoirs of Vandermonde and Laplace are referred to. The only 
fresh and real point of interest lies in the fact that the first index of 
every pair of indices is not attached to the same letter as the second 
index, but belongs to an operational symbol preceding this letter, 
and is used for the purpose of denoting repetition of the operation. 
This and the allied fact that the elements are not all independent 
of each other, A : X 3 and A 2 X 15 for example, being connected by the 
equation 
A 2 X x = A(A 1 X 1 ), 
indicate that Wronski’s combinatory sums form a special class with 
properties peculiar to themselves. 
BINET (November 1812). 
[Memoire sur un systeme de formules analytiques, et leur applica- 
tion a des considerations g^ometriques. Journ. cle VEc. Polyt ., 
ix. cah. 16, pp. 280-302, . . .] 
It would seem as if the above-noted frequent recurrence of 
functions of the same kind had led Binet to a special study of them. 
In the memoir we have now come to, his standpoint towards them 
is changed. They are viewed as functions having a history : for 
information regarding them, the writings of Vandermonde, Laplace, 
Lagrange, and Gauss are referred to: they are spoken of by Laplace’s 
name for them, resultantes d deux lettres , ci trois lettres , d quatre 
lettres , &c . ; and the first twenty-three pages of the memoir are 
devoted expressly to establishing new theorems regarding them. 
Of these the fundamental, and by far the most notable, is the 
afterwards well-known multiplication-theorem. It is enunciated at 
the outset as follows : — 
