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

22 HISTORY OF THE THEORY OF DETERMINANTS

where, firstly, 2(12_)(1d 3 ) is the typical product of two conplementary minors: secondly, the accessories to the E imply that
for B/i, /2 are to be taken any two integers from 1, 2, 3, 4, 5; and,
thirdly, the remaining items are connected with the determination
of sign.
In the way of new contributions the first point to be noted is
that, whereas preceding writers had dealt with the 'extension'
of only special cases of Laplace's expansion-theorem, Reiss gave
the full generalization of the theorem. For example, while Desnanot showed that the identities
ab2c3d4,... = a, bc3d4... - a2 bc3d4... +..,
0 = lab2 la3b41 - I ab3 lla b4l + Ialb4la2b3 I,
could be 'extended,' Reiss affirmed and sought to show that the
theorem
/( a2.. A _. *. a.a.. a
"al a2 ~n' X /i * e * /~A.lI * *\/-,+, A /t
al...,
was a mere preliminary to
l** '*',
/a *.. a jl t. *. i \/ * * * tn X Gil * -lt * *. m \ / -tl * * *. a, t,....t?
(ai...aTi.<n)(I 1) Z1 \ =. 7 T
a,. * a* l
and that the like held in regard to all vanishing aggregates of
products of pairs of determinants. He also takes the further step
of 'extending' that form of Laplace's expansion-theorem which
gives (a1 2 * as an aggregate of products of more than two
\ 21 a n... Cn/
minors, affirming that if the number of minors in each product
be r the left-hand member of the identity is
(, a. an t... t t * t
a1 a2... n Ti Tm... M
More important, however, is the fact that having set himself to
inquire what the left-hand member of Laplace's expansion-theorem
would become if each first factor in the right-hand member were