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

DETERMINANTS IN GENERAL (GORDAN, 18 73) 4. 49

In illustration of one, of Gordan's applications we may note that
if a, ~3, y be the roots of the equation x.3~+cJx2~c+c9_ 3 = 0,1 then
1 C1, C., and a3 a2 a1
/3~ 82 $/31
y3 y2 y
are two arrays conditioned as in the theorem, and therefore that
1 _ -~C1 C2 -C5
I a2$31y0 I - ~3a10yO j a31832_y0- a/3 I (2y
-one of the oldest properties of alternants. This suggests, too,
-that the solution of a set of simultaneous linear equations is involved in the theorem; for example, the satisfying arrays being
a., 2 a3 a4 x y z w
b, b2 b3 b4
C1 C2 C3 C4
we obtain
I(a)b3C41 I a1b3c 4 calb2C 4 - abc
WISSELINK, D. -B. (1873).*
[lETs OVER DETERMINANTEN. 38 pp. Deventer.]
A simply and clearly written introduction well-printed on a large
broad-margined page. It is the first instance-not a translationof the subject being dealt with in the language of Holland.
DOLP, HI. (1873): REIDT, F. (1874).
[DIE DETERMINANTEN, nebst Anwendung auf die L~sung algebraischer und analytisch-geometrischer Aufgaben. Elementar
behandelt. iv+94 pp. Darmstadt.]
[VORSOHULE DER THEORIE DER DETERMINANTEN ffur Gymnasien
und llealschulen. vi+66 pp. Leipzig.]
These are workmanlike elementary class-books. The first is a
particularly clear exposition, higher-pitched in its style than Hattendorif's: the second is a trifle more elementary even than the latter,
*No date on title-page.

M.D. 111.

D