1888 - 89 .] Dr T. Muir on the Theory of Determinants . 
231 
quibus efficiatur 
VlVl + V 2 V2 + •••• + VnVn = + Xfa + . . . . + X n X n1 
simulque data functio homogenea secundi ordinis variabilium 
x 1 ,x 2 , . . . , x n transformetur in aliam variabilium y v y 2 , . . . , ?/ w , 
de qua binarum producta evanuerunt.” 
This being the case he introduces determinants at the outset, 
fixing upon a notation which is practically Cauchy’s, and imme- 
diately using properties of them without proof. Much that is 
contained in the memoir falls to be considered later, as it concerns 
special forms of determinants, — those afterwards known as Jacobians, 
axisymmetric determinants, and, of course, determinants of an 
orthogonal substitution. Indeed, the half-page of introduction is 
almost all that is of interest at present, but even in this a new and 
important theorem is enunciated. The first sentence of it stands as 
follows : — 
“ Supponamus, designantibus a k (m) datas quantitates quaslibet, 
ex n sequationibus linearibus propositis huiusmodi 
y m = a 1 (m) x 1 + a 2 {m) X 2 + .... + a n {m] X n , 
per notas regulas resolutionis algebraicse haberi sequationes 
formse : 
A ^ = ftV 1 + A'V 2 + • • • • +A ( %. 
Ipsum A supponimus denominatorem communem valorum 
incognitarum, qui per algorithmos notos formatur : sive fit 
A = %± a/a 2 " .... eft 
signo summatorio amplectente terminos omnes, qui indicibus 
aut inferioribus aut superioribus omnimodis permutatis pro- 
veniunt ; signis eorurn alternantibus secundum notam regulam, 
quam ita enunciare licet, ut termino cuilibet per certam 
permutationem indicum orto idem signum tribuatur, quo 
afficitur productum sequens conflatum e clifferentiis numerorum 
1. 2 
(2 - 1)(3 - 1) .... (n - 1) . (3 - 2)(4 - 2) .... (n - 2) . (4 - 3) etc., 
eadem numerorum \ permutatione facta.” 
