1888-89.] Dr T. Muir on the Theory of Determinants. 7 57 
the solution of a set of simultaneous equations. Viewing the 
equations 
a x x + \y + c x z = A 
a^x + b 2 y + c 2 z = y > 
a§x + b^y + c^z = £ J 
as giving each of the three variables £, y, £, in terms of the other 
three a?, y, z, we see that on solving for x, y, z , we obtain a con- 
verse system, that is to say, a system giving each' of the three 
x , y, 0 , in terms of y, £. The latter system is, as we know, 
A, , -A.o -A-Qc, 
X= A (+ A 71 + 
B, . B„ B,. 
y=^+^v + -ft. 
Ci> , C 2 , c 3? 
z= -£ £+ a’ + 
where A is the determinant of the original system, and 
-A-l> ®1> Ql» * * * * 
are the cofactors in A of a v b v c v a 2 , .... f respectively. Multi- 
plying the determinants of the two systems, we obtain the determi- 
nant of the quantities 
1 0 0 
0 10 
0 0 1 . 
Hence (p. 176) : — - 
“ Si, n variables 
#, y, z, ... ,t, 
etant liees k n autres variables 
x, y, z, . . . , t, 
par n equations lineaires, on suppose les unes exprime es en 
fonctions lineaires des autres, et reciproquement ; les deux 
r^sultantes formees avec les coefficients que renfermeront ces 
fonctions lineaires dans les deux hypotheses, offriront un 
produit equivalent a l’unite.” (xxi. 6) 
VOL. xvi. 21/1/90 3 c 
