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

108 HISTORY OF THE THEORY OF DETERMINANTS

sion being la1 a 22... a n and b| b22... bn I, or, say, A and B,
he forms the determinant
a l a(12 a1,-... * a bll b1 2... bln,
a21 a22. C a2, n-,i.. a2n, b21 b21 22... b2n................. -.....
anil an2... a, n-m n-rn* * ann bnl bn 2. *. bln
al, 1-m+l al, n-mn+2... al, n-1 a1, bll b12.. bin..... a2, n-m+l a2, n-mr 2..*. a2, n-1 a2n b21 b22 *. b2n
an, n-,n+l an, n-m+2.... an,n - ann bn bn2 *. bn
which, he says, is seen to vanish on trying to find Laplace's
expansion of it in terms of minors formed from the last n+1
columns and the minors that are complementary of those.
Then, noting that the like outcome is not met with when
the boundary-line necessary for the application of the said
expansion-theorem is horizontal and bisects the determinant,
he sets about obtaining the terms of the latter development in
orderly fashion. Clearly, the first factors of those terms are
all alike as regards their first n-Qn columns, but the remaining
im columns may include another column of a's or may not.
Making a separation of the terms in accordance with this distinction, and calling the one aggregate Em-i and the other Em
where the suffix corresponds with the number of columns of b's
appearing in each first factor, and therefore also with the
number of columns of a's appearing in each second factor,
Sperling gives evidence that E,,_ is Sylvester's expansion for
I Ca1 22... ann I I bl22... bnn I when in the formation of it there
is an interchange of nm-1 columns, that Em is the corresponding
expansion due to an interchange of mn columns, and that the two
I's occur with different signs. The conclusion is thus reached
that, if we have previously proved the identity AB = Em-l, the
identity AB = Em must follow.
It is important to note in passing that if Sperling had put
zeros for a1n, a2,.., ann in the second half of his 2n-line
determinant, its value then would have been, when obtained
in one way, (-1)mlAB, and in another, (-1)m-lE,,. He would
thus have made the natural extension of Cayley's simple proof.