The theory of determinants in the historical order of development, by Sir Thomas Muir.

350 HISTORY OF THE THEORY OF DETERMINANTS

the non-zero member of the other equation and taking the product
of the resulting expressions. If in connection with the latter we
make use of Spottiswoode's determinant expression (Hist., ii. p. 111)
for such a non-zero member, the identity evolved will'be purely and
almost alarmingly determinantal.
BALTZER, R. (1864, 1870, 1875).
[THEORIE UND ANWENDUNG DER DETERMINANTEN,. 2te Aufi.
3te Aufi. 4te Aufi. Leipzig.]
Putting (~ 11. 4)
A(x) aox",+aaxm-l+.. +a = a0(x-cj)(x-a )... (x -am),
B(x) boxn +b Xn-1 +.. = b0(X-31)2(x-3)... (X.
and supposing x to be one of the roots of the equation B(x) = 0,
Baltzer predicates the n equations
0 { a,- A (x) + am-ix + am-2X +.
X~~~~~~~~2 +0= l — A(x)'x + Ct-iXL
0z iam-t A(x)}X2..
and the nz equations
0 = 1)+ + b,1x + b,2 x2 ~.
0 bnx + bn1x 2 +.
0.= bx 2 +.
and so deduces
a A m) A(x).-Wt2
A,,x-A(x).
bn~~~~~~~~~
b,~ ~ 0
n+m
which must thus be the equation in A(x) whose roots are
A 31), A( 2),..., A On4 Since the coefficient of the highest,
power of A(x) in it is (- 1)nb 0r, it follows that