1887 .] Dr T. Muir on the Theory of Determinants. 
511 
the development of a determinant according to binary products of 
a row and column. The special row here used is the w th and the 
special column the n th likewise. 
The four pages regarding the application of determinants to the 
solution of a set of simultaneous equations may be passed over with 
the remark that they give evidence of the importance attached by 
Cauchy to his new definition of determinants, the solution in the 
case of the example 
+ b l x 1 = m Y 
a 2 x 1 + b 2 x 2 = m 2 
being first put in the form 
mb(b - m) am(m - a) 
X ~ ab(b - a ) » ^ ~ ab{b - a) 1 
and similarly in the case of the example 
a r x x + b r x 2 + c r x 3 = m r (r = 1 , 2, 3) . 
The determinant solution of a set of simultaneous equations is 
put to good use by Cauchy to obtain new properties of the functions. 
Taking the set of equations 
f + «i- 2^2 *t" d" ay n X n = 771^ 
a + a 2 . 2 x 2 + + a 2 . n x n = m 2 
&c 
a n . -yX\ "I - a n . 2 x 2 “l - ..... *1“ ct n . n x n 
and solving for x v x 2 , . . . he obtains of course the set 
m l b vl + m 2 b 2 . 1 + + m n b n . x = D n x l: j 
mf v2 -f mjb 2 . 2 + + m n b n . 2 = D v x 2 , j 
&c 
mfjy n + m 2 b 2 . n + + m n b n . n — X) n x n , J 
where b ri , b 2 . l , have the signification above indicated, 
and D n stands for S( ± a vl a 2 . 2 . . . a n . n ). This second set may 
be treated in the same way as the first set, the quantities 
m v m 2 , . . . , m n being viewed as the unknowns. To express the 
result the system of quantities adjugate to (b rn ) is denoted by 
(c 1<n ), and the determinant of the system (b Vn ) is denoted by B M , the 
new set thus being 
vol. xiv, 13/2/88 2 R 
