1888 - 89 .] Dr T. Muir on the Theory of Determinants. 761 
Here then we have for the first time in the notation of deter- 
minants the pair of upright lines so familiar in all the later work. 
The introduction of them marks an epoch in the history, so im- 
portant to the mathematician is this apparently trivial matter of 
notation. By means of them every determinant became represent- 
able, no matter how heterogeneous or complicated its elements 
might he; and the most disguised member of the family could he 
exhibited in its true lineaments. While the common characteristic of 
previous notations is their ability to represent the determinant of 
such a system as 
a 1 ^2 
% 
^1.1 a \,2 tt l,3 
\ or 
a 2,\ a 2,2 a 2.3 
c 1 c 2 
C 3 
a 3,l a 3,2 
and failure to represent 
in the. case of systems like 
a b c 
a b 
c 
4 5 6 
cab 
1 a 
b 
3 2 7 
b C Or, 
0 1 
a, 
8 10: 
Cayley’s notation is equally suitable for all. To illustrate by 
analogy, — the infinitesimal calculus supplied with Lagrange’s nota- 
tion for the differential coefficient of cf>(x), but unable to symbolise 
the differential coefficients of such a special function as ax 3 + bx 2 , 
or log (1 — a?)/(l + 3?) would be in the exact predicament of the 
theory of deteminants prior to Cayley. 
Of less importance is the fact, which the quotation indicates, that 
Cayley had discovered for himself the multiplication-theorem, but 
characteristically hesitated to proclaim it new : also, that, probably 
following Yandermonde, he took the recurrent law of formation for 
his definition, making the signs all + in one case and + and - 
alternately in the next, exactly as Yandermonde did. 
He then proceeds to the seemingly geometrical problem : — 
“ To find the relation that exists between the distances of 
five points in space. 
“ We have, in general, whatever aq, y lr z v &c., denote, 
