1887.] Dr T. Muir on the Theory of Determinants. 
505 
be performed , and in the result every index of a power be changed 
into a second suffix, e.g., a r s into a r>s , the expression so obtained is 
called a determinant (vm. 2), 
and is denoted by 
S (±a 1 . 1 a 2 . 2 a 3 .% .... a n . n ) (vii. 5). 
In this definition the rule of signs and the rule of term-formation 
are inseparable — a peculiarity already observed in the case of 
Bezout’s rule of 1764. 
After the definitions various technical terms are introduced. The 
n 2 different quantities involved in 
S ( + • • • • ®rvn) 
are arranged thus 
f a ri 5 
a V2 1 
a l‘3 ’ * * 
. . a^. n 
j a 2‘l ’ 
a 2'2 "> 
a 2- 3 ’ ‘ • 
. . a 2 . n 
] a 3’l ’ 
&c. . 
a 3'2 J 
CO 
CO 
e 
. . a^. n 
. a n’l 5 
a n . 2 , 
a v .% , . . 
• ° a n . n 
“ sur un nomhre egal a n de lignes horizontales et sur autant de 
colonnes verticales,” and as thus arranged are said to form a 
symmetric system of order n. The individual quantities a vl , &c. 
are called the terms of the system, and the letter a when free of 
suffixes the characteristic . The u terms ” in a horizontal line are 
said to form a suite horizontal, in a vertical column a suite verticale. 
Conjugate terms are defined as those whose suffixes (“ indices ”) 
differ in order, e.g., a 2 . 3 and a 3 . 2 ; and terms which are self-conjugate, 
e.g., a vl , a 2 . 2 , . . . are called principal terms. The determinant is 
said to belong to the system, or to be the determinant of the system. 
The parts of the expanded determinant which are connected by the 
signs + and - are called symmetric products, and the product 
a vi a 2’2 a z'z • • • • a n . n 
of the principal “ terms ” is called the principal product. The 
“ principal product,” however, is also called the terme indicatif of 
the determinant, and thus an awkward double use of the word 
“ terme ” is brought into prominence. The system 
