1907-8.] 
Dr Muir on General Determinants. 
683 
Spottiswoode, W. (1853). 
[Elementary theorems relating to determinants. Second edition, 
rewritten and much enlarged by the author. Grelles 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 Deter- 
minants,” the following appears (p. 232) for the first time : — Theorem ix. 
The sum of two determinants in which i rows {on a certain level) are 
respectively equal, is equal to the determinant whose i th minors on the 
aforesaid level are identical with the corresponding \ th minors of each of 
the two given determinants, and whose (n — i) tt complementary minors are 
respectively the sum of the complementary minors of the given deter- 
minants. 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 Sylvesters theorem of 1852 (October). Spottis- 
woode’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 
a 
b 
c 
a 
P 
7 | I 
aa 
ad 
a a" 
b 
c 
a! 
1) 
c' 
d 
P 
y 
a a 
ad 
ad' 
b\ 
c 
a 
b" 
c 
a" 
F 
y" 1 
Ct" a 
a' a 
rr rr 
a a 
b" 
rr 
C 
P 
p 
13- 
y 
y 
y" 
essentially consists in expanding the right-hand determinant in terms of 
minors formed from the first three rows and minors formed from the last 
two rows. His other fresh proof is dependent on the connection between 
determinants and simultaneous linear equations. Taking the two sets of 
equations 
ax + ay + d'z = «q | au x 4 - bu 2 + cu z = v Y J 
fix + fly + = u 2 V a'u x + b'u 2 -1- c'u z = v 2 ?- 
yx + y'y + y"z = u 3 J d \ q -f b"u 2 + c"u 3 = v 3 j 
and substituting for u v u 2 , u 3 , in the second set and solving, there is 
obtained for x an expression whose denominator is known to be 
