Dr T. Muir on the Theory of Determinants. 
527 
minants. The treatise is logically arranged and carefull}^ written. 
It opens with an introduction of 4 pp., the main part of which 
serves as a table of contents and as a guide to the theorems which 
the author considered his own. It consists of four Sections 
(Abtheilungen), subdivided into portions which we may call 
chapters, the first section containing five chapters, the second also 
five, the third one, and the fourth four. 
Schweins’ name for the functions is 
Producte mit Versetzimgen ; (xv. 6.) 
his notation is a modification of Laplace’s, viz., he uses 
) (vii. 6) 
where Laplace used simply 
( 
and his definition is the same as Vandermonde’s ; that is to say, he 
employs Bezout’s law of recurring formation. His words at the out- 
set are — 
“Die Bildungsweise der Producte, welche hier untersucht 
werden sollen, geben folgende Zeichen an : — 
1 ^2 \ II \ ^2 
L ^2/ “11 ^1/ • 
1 32 ^3 \ _ |l 3i ao \ 
‘1 -^2 ^3/ I ^1 “^2/ ’ 
32 \ 3i 
•^1/ • “^2 J 
3 i as \ 3-2 II a 2 ao \ ai 
^2^ * ^3 II ^1 ^2/ • ^3 > 
82 as 84 \ II 84 ao 83 \ 84 || 84 ao 84 \ ‘83 
Ai A .2 A3 A4J II Aj A'2 AgJ • A4 II Aj A2 AgJ • A^ 
II 34 83 84 \ 82 II 82 33 84 \ 84 
II "^2 ■^3/ ‘ -^4 II ■'^1 -^2 -^3 / ' “^4 7 
und allgemein 
|a,....aJ = (-)||a,.... A,,J. A,. + (-) ||a,....a„Ja„ 
, / \*|1 • -^ 3-^-1 ^n-x+i . . 3 „ \ a„_ 3 j I a 2 a,, \ aj 
+ a,.Ja„ +•••+(-) |aj..,a,Ja„ 
Oder 
|| 3 „ \ . V j| 84 . . . a„_a;_i an-x+a • • • a,, \ ^n-x rv 1 -1 
||Ai....aJ= ||Ai a„J.a„ *=o,i, . . . , »-i. 
