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

408 HISTORY OF THE THEORY OF DETERMINANTS

suggests the finding (pp. 80, 81) of the results
1 b.....
-1 1-b b.... =
- 1-b2......... * n+1
and
1 —b1 b..... = 1-b + bb2- blb2b3 +
— 1 1-b2 b3 *.. -1 1-b3............... n,
the latter of which may be compared with a result obtained by
Nachreiner when under the same influence.
GUNTHER, S. (1873).
[Ueber die allgemeine Auflosung von Gleichungen durch Kettenbriiche. Math. Annalen, vii. pp. 262-268.]
The starting-point here is Fiirstenau's expression of 1860 for the
smallest root of the equation
Xn -+ a,_Xn-1 +.. + a2x +ax + ao = 0.
All that is of interest in the paper from our present point of view
is the fact that Fiirstenau's two determinants are changed so as to
have the zero elements on the under side of the main diagonal
matched by like elements in the corresponding places on the upper
side, with a view to expressing the root of the equation in the form
of an interminate continued-fraction. The procedure need not be
described at length; it will be understood at once by applying it
in the case of the determinant
a1 a2 a3 1
a0 a1 a2 a3. a,
-a course which also enables us to test the accuracy of the illustrative example given later in the paper. In the first place, the
element in the place (1, 4) is changed, the result being