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

186 HISTORY OF THE THEORY OF DETERMINANTS

and that the said function becomes homogeneous on multiplication by a2-2. The actual result is not given, but in the second
edition (1864) it is stated to be
an an-1 n- 2....
7.(2X1-2
- * C =.............
0an ~l ( ) - 1 (-2)(G_2....
aCn- 2an_2 3-_....
"eine Determinante (2n-2)ten Grades, bei welcher die m-2
ersten und die mn-l folgenden Zeilen in Bezug auf die nicht
verschwindenden Elemente iibereinstimmen."
Part of the object which Baltzer had here in view was to
establish the relation between two forms of the discriminant of
the given equation; namely, that obtained by squaring the determinant-form of the difference-product and that obtained as
the eliminant of the equations
f'x) = 0, Xf(x)-xf'(x) = 0,
or the equations
(cx"+ca - xl ty... +cty) = 0
-(aC, xn+ay + t... o+ a0) = 0
'y
Now a glance at the final determinant suffices to show that it is
not the eliminant sought, there being in it three types of rows,
whereas the two equations giving rise to the said eliminant being
both of the ( - l)th degree, the coefficients of the one must occur
in as many rows of the eliminant as the coefficients of the other.
Further, since the coefficients of the equation f'(x) = 0 are seen
to occur in their full number of rows, and those of the other
equation in the last row only, it is therefore the first n -2 rows
that need to be changed. The set of operations requisite to
effect this is