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

140 HISTORY OF THE THEORY OF DETERMINANTS

BRIOSCHI, F. (1854,; March).
[LA TEORICA DEI DETERMINANTI, E LE SUE PRINCIPALI APPLICAZIONI. viii - 11 6 pp. Pavia.]
Unlike Spottiswoode, IBrioschi in methodical manner defines
"un determinante simmetrico," and gives four known properties
expressed in clear language, all within the space of one page
(P. 70).
BRIOSCHI, F. (1854, Dec-.; 1855, Apr.).
[Sur quelques questions de la geoinetrie de position. Crelle's
Jomtbrnal, 1. pp. 233-288; or Opei-e inat., v. pp. 259-265.]
[Relations de distance entre des points. Nombv. Annales de
MaLth., xiv. pp. 172' 173; or Opere mnat., v, pp. 123-124.]
The first title here recalls that of Cayley's maiden effort;
and, as a matter of fact, the paper of 1854 had its origin in the
paper of 1841. Cayley, it will be remembered, obtained the
relation connecting the mutual distances of five points in space
by multiplying the two determinants
Z -2x1 -2yi -2z1 -2w1 1 1 1 t4 01
Ix22 -2x -2Y2 -202 - 2W2 1 1 X2 Y2 02 lW2 IX22
5X -2x5 -2y5 -205 -2w 1 1 X5 Y5 -5 W5 X52
1...... 1,
the first of which is -16 times the second, and then putting the
w's equal to zero. Brioschi now follows on the same lines,
but with an interesting difference. Having shown that the
determinant
x12++y12~ols2 (X6 -X1)2+(y6-y1)2+(o6 - o )2 1 - 2x, - 2Yi - 2z,
x22~+y 922 + Z22 (X6-X2)2+(y-y2)2+(Z6 - 2)2 i - 2X2 - 2Y - 22
X2y2+ Z5 (x6-X5)2+(y6-5)2~(o5-z)2 1 - 2X5 -2;/ -225
1 1 t b
vanishes identically, the simple fact being that the second