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

368 HISTORY OF THE THEORY OF DETERMINANTS

This is what Pasch accomplishes. He begins with the much
more general identity
tull U12.. U*ln E(y lr)
'U21 U22... 2n u(Yu 2?r)
=0,
UnI U.... U..(..ur)
V1 V2.. n
and derives therefrom three others by operating thrice with
'DX, aX2 an
S1 + + D Xn
From this set of four he obtains two other sets by making the
substitutions
Y1, Y2, * * n =, X, -,* * * n,
Y1, Y2, * * *, Yn = 1, CL, 2 * * * n n;
and then by combining the first, second, third of the third set with
the second, third, fourth of the second set he reaches his general
results. To bring these into close touch with Hesse's proposition
it only remains to put
V1n, 2, ' ' * n = Uln U2,... ~,n.
The paper closes with matter analogous to Gundelfinger's, namely,
with the enunciation of the conditions for a ternary cubic being
Tesolvable in various special ways: the quaternary cubic is, however, also now dealt with.
GORDAN, P. (1875): NOETHER, M. (1876).
[Ueber einen Satz von Hesse. Sitzungsb.... Soc. zu Erlangen,
viii. pp. 89-94.]
[Ueber die algebraischen Formen mit identisch verschwindender
Hesse'sche Determinante. Sitzungsb.... Soc. zu Erlangen,
viii. pp. 51-56.]
[Ueber die algebraischen Formen, deren Hesse'sche Determinante
identisch verschwindet. Math. Annalen, x. pp. 547-568.]
In the first of these papers Gordan, going a little farther than
Pasch, shows that Hesse's contested proposition holds for all ternary