1888-89.] Dr T. Muir on the Theory of Determinants. 
207 
The Theory of Determinants in the Historical Order 
of its Development. By Thomas Muir, M.A., LL.D. 
Part I. Determinants in General (1829-35). 
(Continued from p. 544 of vol. xv.) 
KEISS (1829). 
[Memoire sur les fonctions semblables de plusieurs groupes d 7 un 
certain nombre de fonctions ou elemens. Correspondanee 
math, et phys., v. pp. 201-215.] 
In Beiss we have an author who starts to his subject as if it 
were entirely new, the only preceding mathematician whom he 
mentions being Lagrange. Like Cauchy he opens by explaining 
a mode of forming functions more general than those of which he 
afterwards treats, the essence of it being that an expression involv- 
ing several of the n.v quantities, 
a a 
at 
ay 
. . . aP 
b« 
W 
by 
. . . bp 
c a 
c/3 
cy 
... CP 
r°- 
n 3 
ry 
... rp 
is taken, and each exponent (“exposant”) changed successively 
with all the other exponents, a, or each base changed 
with all the other bases, a, b, ... . Only a line or two, however, 
is given to this, the special class known to us as determinants 
being taken up at once. 
His notation for 
a l b^c s - a x 6 3 c 2 — a 2 6 1 c 3 + a^c 1 4- a^b 1 ^ - a^b 2 c^ 
is 
(abc , 123) , (vn. 7) 
a line being drawn above the exponents to indicate permutation. 
His rule of formation of the terms and rule of signs are combined 
after the manner of Hindenburg. Like Hindenburg, he arranges 
the permutations as one arranges numbers in increasing order of 
magnitude ; but, unlike Hindenburg, after the arrangement has 
