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

138 HISTORY OF THE THEORY OF DETERMINANTS

To these may be added from another section (~ 3. 15) the result
- ao... -1 ( - 1), - ( a) ao
______aal 1 ~ ~ ~ an-1 -
STERN, M. A. (1865).
[Ueber einen Satz aus der Determinantentheorie. Crelle's Journ.,
lxvi. pp. 285-288.]
The theorem is that if pl, 02,.. be functions of the first, second,... degree in the variable, and have 1 for the coefficient of the
highest power in each, say
01)(x) = X + 11
02(x) = x2 a 2x- -a2,
-(x) = x-l+a x-+...+*- -, - X
then the determinant
1 p1(xI) Q02 ().. -1(X1)
1 k1(X2) 2 (X2).. * 1~(X2)
1 C(1(:n) 2 (X,)... 0(X-1(Xn)
is equal to the difference product of xl, x,..., x, being in fact,
although Stern does not say so, the difference-product multiplied
cow-wise by 1 in the form
I1.
1al......
a,1 1......
a29 a12 1.
a33 a23 3 1.* *
The theorem is clearly a case of that which we have spoken of under
Baltzer (1864) as a "so-called theorem." By means of it the
determinant
| CZO C,,0 0,: C3,2 * * * O.'-i1 |In