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

408 HISTORY OF THE THEORY OF DETERMINANTS
and obtains a product-determinant from whose columns, he says,
the factors 01, 02,..., may be removed in order, so that
there results
1 1 1... 1
0,,, 1 2 n-i
1A a, 1 2 *a-1
DA = 0102 On 1-2 a,-.2 - a n, -2
1 a1 a2 a.
= 1... 0.n (- 1 ) -)
and.. D = (-1)("-. 012... n.
The proof, which is said to be due to Brioschi, is not improved in
neatness by introducing the conception of a primitive root, nor by
writing the root 1 in a different form from the other roots.
The second lemma concerns the differential-quotient of D with
respect to any variable of which the a's are functions. Denoting
this differential-quotient by D', and by D, the determinant got
from D by substituting for each element in the rth column the
differential-quotient of that element, Cremona of course has
at once
D' = D,+D2+...+D,.
As, however, D1 here can be shown by translation of a number of
rows and the same number of columns to be equal to any one of
the D's following it, there results
D' = D n = nD....
The third lemma is to the effect that the quotient of the
determinant
m qo q1d.... qn-_2d-2
mid qld q2d2.... q,,-idj-1
mn_ Idt-li ql-ld -l qo.... q-, dn-3
by d is a rational function of dn. By multiplying the 2nd, 3rd,
4th.. columns by d", dn-l, dn -2,.... respectively, and then