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

198 HISTORY OF THE THEORY OF DETERMINANTS

be denoted by BB, B2..., B; then
A1.B1. A. B2... A1.B1
A2 B1 A2B2... A2'.BX = (A - B)C?^1
A. B1 Al.B2... At.B1
By way of proof, Sylvester merely states that it is obtainable
from his general theorem of March 1851, "by making
an,,+I am,+2 '. n+n (+
bm+l bn+2..* br+n
represent a determinant all whose terms (i.e. elements) are zeros
except those which lie in one of the diagonals, these latter being
all units."
His only other remark is that when r = 1 and when r= gm the
right-hand members are identical, and that the equating of the
two left-hand members which is thus legitimised gives Cauchy's
extended multiplication-theorem.
IThe former remark, unfortunately, is another troublesome
instance of inaccuracy. The specialisation given therein is only
one of two which are needed, the other being that every element
of the determinant
a1 62 ( '3. aC,
1 b2 b3... b,
be made 0. To make the matter clear, let us take the case of
the general theorem where m= -4, n= 5, r =2, and perform the
requisite specialisations. The general theorem then is
a a2 5 a..1. 2 0.5 3 9. a9 a2 a5 a,..
1 &2 Ct5 7 * * * 9 | b 7,C 5bf. ^.. l r c5 6 ' ' ' {9 g
b b1 b25 o b. b bb... I b b4 b, b5.... b9
a a3 5 a 6...* a.9 a, U 3 (5 6 * * * a59. ( 6. *.
i bai a 5... 1 Q | ah 3 b)Cf5,.. 1. 1 ), | 4 1) b *. * b 9
jt Lit3 Co5 a 6.. Ct9 L 1 ( 3 0 * *. 9 83 C4 a 5 a 6 * * 9 0
1 b bs.. | b b b l.. ): b b bI b... b b9
~ ll. / 0, li4.a.. 9 a~ 3i a... (9
- g Lt 5L6,..09 t6" a *1a... o9
b be...b9 b1b2... b9