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

BIGRADIENTS (TRUDI, 1862).331

is evidently n: thus, in the case where rn,n = 7,4 the arrays are
(a1,.. (a,a7)3 (a0,... a7)4 I
(b0,...,,4) (b0,...,b4)5,, (b0,. b4)6 K (boy. b.. 7
the last, where r = n, being square, and therefore preferably
written in the form
(boy,.. b,
These and other preliminaries being settled, he is in a position
to deal with an important theorem on the subject of what we may
call the condensation of a bigradient array. The proof given is,
unfortunately, not at all so simple as it might have been. We
shall therefore substitute for it one of our own, which Trudi himself
would probably have devised had he been aware of Cayley's work
of 1845. Taking, first, a case in which m = n, say the case

I aoo
II.
11 b

al a2 a3 a4
ao al a2 a3 a4
ao al a2 a3 a4
bo b1 bg b3 b4
bo b1 b2 b3 b4
b1 b2 b3 b4

i (a0,
or 1!
iii(bo),

*, a4)3 H
*, b 4)3,

we multiply the determinant

1

1

- bo
1 -bo -bi
* a0 a1

- bo
- bi
a2
a1
ao

by the given array in column-by-column fashion, obtaining (Hist.,
- QA\

11. PI 3YI)

(a0,.., a4)3 -
(bo,... -, b4)

ao at a, a, a
a a1 a2 a3 a4
a0 al!- a3 a4
a0 al a2 a. a4
aob1 1aob21I laob3l Ja~b41
i.. jla,b2j ja~b3 + ja1b2 I Jab4j +la~b3l aabb 4
*.. zJaob3 Iaob41 + aba I JIalb4 +la Al Ja2b 4