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

206 HISTORY OF THE THEORY OF DETERMINANTS

so that, on using again our initial identity, we have
YIZJ Z2X,5 X3Y611= f")Z ccyzHX4Y5Z6KI~yz1 z2X.I 1X3Y1Il
which is the result sought to be established.
As the said initial identity is not essentially different from one
given by Scholtz in the second volume of MLiiegyeterni Lapok, it is
important to note that Ilunyady here states that the substance of
the present paper had appeared previously in the first volume of
the same publication.
PASCH, M. (1879).
[Ueber gewisse Determinanten, weiche in der Lehre von den Kegelschnitten vorkommen. Crelle's Journ., lxxxix. pp. 247-251.]
Save in one instance the determinants referred to in the title
are those with which Hunyady, Mertens and Scholtz have already
made us familiar. The demonstrations are Pasch's own, but
present no feature of particnlar interest. The fresh result is
X2Y3Z5 X2Y3Z6 X9j 3Z6 X2Y(Z4 x.2y2qz4 X2Y4 [
I J I X3Y1XO YZ6 X3YIZ4 i I 4 X3YIZ5
i X1Y2z 5 I Y2z6 X1Y2Z16 X1y~z4 i I X1Y2XCYZ4 X1yJz5
I XIY2Z3 12 I Xi2 Y22 -32 Y4z4 Z5x5 X4Y6 f.
By expanding the left-hand member in terms of the elements of
its first row and their complementaries, we obtain without difficulty
(235.236.314.124.165 - 236'234-315 125-164 + 234-235.316-126.154) 123.
But of'the three terms here within the brackets,
the first = 235-236-314(125-416-126-415),
the second = 236-125-146(235-431-231A435),
the third = 235-126-154(236-314-231-346),
and consequently their sum is
-(236-125-146-435 -235-126*154-346) 123.
We thus have for our compound determinant the equivalent
1232(125.146.236.345 -126.145.235.346),