1904 - 5 .] 
Dr Muir on General Determinants. 
925 
The preface contains a short sketch of the history of the theory, 
the first of the kind that had appeared. In the first section (§ 1) 
the reader is introduced to determinants of the second and third 
orders as they actually occur in the solution of geometrical 
problems, and certain of the simpler properties are incidentally 
pointed out; the next seven sections (§§ 2-8) deal with deter- 
minants in general ; and the rest of the hook is occupied with 
determinants of special form, viz., § 9 with skew determinants and 
§ 10 with functional determinants. The concluding portions of 
most of the sections consist of worked examples illustrative of 
applications of the theory to co-ordinate geometry. 
The definition employed is that of Vandermonde * as expressed 
in Cayley’s vertical-line notation, viz. — 
( 1 . 1 ) ( 1 . 2 ) 
( 2 , 1 ) ( 2 , 2 ) 
= ( 1 , 1 )|( 2 , 2 )|-( 1 , 2 )|( 2 , 1 )|, 
(1.1) (1,2) (1,3) 
(2.1) (2,2) (2,3) 
(3.1) (3,2) (3,3) 
-( 1 . 1 ) 
(2.2) (2,3) 
(3.2) (3,3) 
+ ( 1 , 2 ) 
(2.3) (2,1) 
(3.3) (3,1) 
+ (1,3) 
1 ( 2 , 1 ) 
1(3,1) 
( 2 , 
(3, 
The quantities (1, 1), (1, 2), . . . which Cauchy called ‘terms’ 
and Jacobi ‘elements,’ are named ‘constituents' ; and the deter- 
minant of the n th order having these constituents is denoted 
shortly by 
2±(l,l)(2,2...(n,n) 
The first result, deduced in somewhat loose fashion from the 
* It may be worth noting that while both Vandermonde and Schweins 
used the recurrent law of formation as a definition, they did not write it in 
exactly the same form. Schweins followed closely the form used by the 
original discoverer, Bezout, putting for example 
| «A c 3^4 | = d 4 |«A C 3 | ~ ^8 | a lb‘2 C 4 l + d 2 | « A C 4 \ ~ d l\ a 2 6 3 C 4 | , 
the connecting signs being in all cases alternately positive and negative ; 
whereas Vandermonde wrote 
| a^c 3 d 4 | = a 2 1 & 2 c 3 d 4 | - a 2 1 & 3 c 4 ^ 1 j + a ?t | & 4 6' 1 c? 2 ( - a 4 1 & 4 c 2 d 3 | , 
where the cyclical change of suffixes causes the connecting signs to be 
alternately positive and negative when the order of the determinant is even, 
and to be uniformly positive when the order is odd. 
to to 
