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

36 HISTORY OF THE THEORY OF DETERMINANTS

outset he consequently devotes a separate section (~ 2, pp. 530 -533).
The first theorem may be formulated as follows: If two arrays
C and D, each consisting of n-1 rows and n columns, be so related to
one another that in every case the product of the rth and every succeeding
row of C by the rt" row of D vanishes, then the principal minors of
the last k rows of C are proportional to the principal minors of the
first n-k rows of D, the series of minors in the latter case being arranged
in reverse order. Taking for shortness' sake the case where n is 5
and k is 2, the arrays being
al a2..* a5 xl x2... X
b, b2... b5 Y1 Y2.. Y5
C1 C2 ~. C5 Z1 Z2... Z
d1 d2... d5 w W2... w5,
and the given relations therefore representable by
fax = 0,
2bx = O, Eby = 0,
Ecx==, Ecy = O, Ecz = 0,
Idx = O, Edy = O, Edz= O, vdw= 0,
we append 1 row to C and prefix k rows to the first n-k rows of
D, thus obtaining the square arrays whose determinants are
I acb2c3d4e5 1, I mln2X3Y4Z5 j.
By the multiplication-theorem the product of these
2am Ian. Lay xaz
_bm Ibn.. Ibz
= cm C cn
dmn 2dn.
2em 2en 2ex 2ey /ez
= ex * lay * 2bz I cm 2cn
Vdm 2dn
= 2ex * 2ay b * 2bz c1 C2 C3 C4 c5 | I l mI 1 3 m4 i m3 5
d1 d2 d3 d4 d5 n, n2 n3 n4 n5,
from which, by equating the cofactor of hnm, on the one side
to its cofactor on the other, we obtain the desired result.