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

DETERMINANTS IN GENERAL (SPOTTISWOODE, 1853) 81
d2-wd1 d3-wd2 d-wd 3
d3-wd2 d,4- od d5- od, = 0.
d, -wd d5-wd 4 d6-wd5
SPOTTISWOODE, W. (1853).
[Elementary theorems relating to determinants. Second edition,
rewritten and much enlarged by the author. Crelle's
Journal, li. pp. 209-271, 328-381.]
A more correct description of Spottiswoode's second edition
would be rearranged, partly rewritten, and much enlarged, the
majority of the titles of the old sections or chapters occurring
again but in a different order, the majority of the sections being
enlarged, and two or three new sections being inserted. Although
the total increase of matter is from 71 pages to 117, there is comparatively little to be noted concerning general determinants.
In ~2, which bears the title "Addition and Subtraction of
Determinants," the following appears (p. 232) for the first
time:-THEOREM ix. The sum of two determinzants in which i
rows (on a certain level) are respectively equal, is equal to the
determinant whose ith minors on the aforesaid level are identical
with the corresponding it' minors of each of the two given
determinants, and whose (n-i)tl complementary minors are
respectively the sum of the complementary minors of the given
determinants. No instance is given where the two determinants
have more than one row different.
In ~ 4, which deals with the multiplication of determinants,
much space (pp. 238-248) is given to Sylvester's theorem of
1852 (October). Spottiswoode's own mode of treating the subject
is to begin apparently with the two factors and arrive at the
product, whereas in reality the opposite is the case. For
example, his proof that
aCa Ca' caa" b c
a b c a /3 aa a'a' a'a" b' c'
a' b' c'. a ' y' = a"a a"a"' a"c" b" c"
7 7 7 7'
M.D. " I.
M. D. I. F