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

430 HISTORY OF THE THEORY OF DETERMINANTS

while the other is obtained from this by putting x = n-i. Denoting his own form by U,, Cayley, with Sylvester's results
before him, found
UT2 = (02-1) - (X-1),
U3 = O(O0-4) - 3(x-2)O,
U4 - (O2-1)(20_9) - 6(x-3)(02-1) + 3(x-3)(x-1);
so that, if he put Ha for the value of U,, iu Sylvester's case (viz.,
when x= n-1), he could write
U2= H2 (X-1)10
U3 = 11 - 3(x-2)H,
U4 = H4 -6(x-3)H'2~+ 3(xc - 3)(x - 1)Ho,
and thence, doubtless, divined the generalisation
U11 = Hn, - Ba,.'(x-n~+ 1).H,,n, + B,,2.(x-n+ 1)(x -n+3).Hn4....-.
where
I,, = (O+n-l)(O+rb-3)(O+n-5)....toii factors
and
- n(n-12 )(n/-2).....(/-~ -i
n's, 2s.1.2.3.... s
The establishment of the truth of this is all that the paper is
occupied with, the procedure being to expand U,, in terms of the
elements of* its last row and their complementary minors, thus
obtaining
U,, = OU,,n, - (n- l)(x-ni+2)Un,,
and thence
U,, + {((n-1)(Qx-n+2) + (n-2)(x-n+3) - ult+ (~n - 2~n -. )(x - ~n 4- 3)(x - n~ + 4~)Un = 0,
and showing that the above conjectural expression for U, satisfies
the latter equation. The process of verification is troublesome,
and was not viewed with satisfaction by Cayley himself.
As a preliminary the coefficients of the H's in the value of U,,
are for shortness' sake denoted by A,,,, -A,,,1,..., and for