400 Proceedings of Royal Society of Edinburgh. [sess. 
i 
inconnues, pourvu que Ton prenne convenablement le signe du 
num4rateur.” 
He then proceeds — 
“ Snpposons done que pareille verification ait ete faite pour 
n- 1 equations entre n - 1 inconnues, je dis qu’elle se fera 
encore dans le cas de n equations.” 
How although the statement in 2° is true for the case of three equa- 
tions, it is not true generally, and therefore cannot he proved.* 
The theorems which follow this introductory matter concern a 
special determinant, viz., the determinant of the system, 
a Y b 1 c 1 7c x l x 
a 2 ^2 C 2 ^2 ^2 
a n b n c n . ... k n l n , 
in which the elements are connected by the \ n{n- 1) relations 
a A 
+ 
af)^ 
+ 
a A 
+ . . 
, . + 
a n b n 
= (T 
a x c x 
+ 
^2 C 2 
+ 
a s c z 
+ . , 
. . + 
a n c n 
= 0 
Mi 
+ 
a 2^2 
+ 
+ . 
. . + 
a Jn 
= 0 
Mi 
+ 
h C 2 
+ 
^3 C 3 
+ . 
. . + 
brfin 
= 0 
Mi 
+ 
b^d^ 
+ 
Ms 
+ . , 
. . + 
b n d n 
= 0 
Mi 
+ 
b^2 
+ 
+ . . 
. + 
bJn 
= 0 
Mi 
+ 
^2^2 
+ 
+ . . 
. + 
= 0, 
Such determinants are only a little less special than determinants 
of an orthogonal substitution, and thus naturally fa)l to be con- 
sidered later along with those of the latter class. 
4 
* In the proof he is fortunate (or unfortunate) enough to use another 
special case in which the statement is true. He says: — “ Les deux termes 
et e 7 / 6 a 1 & 3 c 5 ^ 2 qui entrent dans D 4 , et qui se deduisent l’un de 
1’ autre par une permutation tournante entre les lettres ont meme signe.” 
