of Edinburgh, Session 1885-86. 567 
(1) A simple and appropriate notation for the new functions, e.g 
1|2|3 . (VII) 
(2) A new mode of defining the functions, viz., using B6zout’s 
recurring law of formation, (vin.) 
(3) The remark that the ordinary algebraical expression of any of 
the functions is obtainable by permutation of either series of 
indices, . (ix.) 
(4) The remark that the positive and negative terms are equal in 
number, (x.) 
(5) The theorem regarding the effect of interchanging two con- 
secutive indices, (xi.) 
(6) The theorem (with proof) regarding the effect of equality of 
two indices belonging to the same series, (xit.) 
(7) A reasoned-out solution of a set of n simultaneous linear equa- 
tions, by means of the new functions as above defined, . (xni.) 
(8) Expression of any of the new functions of order 2m 
as an aggregate of products of like functions of orders 2 and 
2m - 2, (xiv.) 
In addition to this, we must view Vandermonde’s work as a 
whole, and note that he is the first to give a connected exposition 
of the theory, defining the functions apart from their connections 
with other matter, assigning them a notation, and thereafter logically 
developing their properties. After Vandermonde there could be no 
absolute necessity for a renovation or reconstruction on a new basis : 
his successors had only to extend what he had done, and, it might 
be, to perfect certain points of detail. Of the mathematicians whose 
work has thus far been passed in review, the only one fit to be viewed 
as the founder of the theory of determinants is Vandermonde. 
LAPLACE (1772). 
[Recherches sur le calcul integral et sur le systkme du monde. Hist, 
de VAcad. Roy. des Sciences , Ann. 1772, 2 e partie (pp. 267-376) 
pp. 294-304]. 
In the course of his work Laplace arrives at a set of linear equa- 
tions from which n quantities have to be eliminated. This he says 
