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

400 HISTORY OF THE THEORY OF DETERMINANTS

it, and thereafter diminishes each column except the last by the
columns immediately preceding it, and thus obtains
Qr = -n 1..... +l1
-n -a- al 1.... ar + 1
~ - -n — a a...... a 1..... a,,-, ar +1. -n-ari ar+l
and therefore

= -nD +(n+ai) D2-(ar+1) I},
if DI be put for the determinant got by deleting the first row and
first column of Qr, D2 for the determinant got by deleting the first
two rows and first two columns, and II for
(n+-a2)(n+ a3)... (n+ a-_l).
Similarly, of course,
Pr = n -n 1...... +
- n-a2 a 1..... ar +............._1 a, -1........ ar-1 ar +
~....... -n-ar-i ar
and thus, by altering the (1, 1)th element into al- (n+ a,), there is
obtained Pr = nD1 - n(n + a) D2;
so that on the elimination first of D1 and then of D2 from these
expressions for Q, and Pr, there results
Pr + Qr = (n+ai){(1-n)D2-(ar,+1)I},
Pf + nQr = n{(l-n)D-(n+al)(a,+l)ll}.
These, on returning from D1, D2, II to their lengthy equivalents,
are changeable into
Pr + Qr = (n+al) 1- n....... ar
- n-a2 a2 1.... ar

1
1

-n-a3 a3.... 1.... ar12..... - -r —2.... — n — r at_

a, + 1
ar + 1
ar+1
ar