1907-8.] Dr Muir on Compound Determinants. 205 
express the product of two m-by-n arrays as a determinant of the 
(m + nf h order.* 
Spottiswoode (1853). 
[Elementary theorems relating to determinants. Second edition, 
rewritten and much enlarged by the author. Crelles Journal , 
li. pp. 209-271, 328-381.] 
In his first edition Spottiswoode had a section (§ vi.) headed On Inverse 
Systems and Determinants of Determinants ; but in it, as we have seen, 
he dealt merely with the adjugate determinant and its minors. Now, this 
is supplanted by a section (§ x.) of considerably greater extent (pp. 350-372) 
with the short Sylvestrian title On Compound Determinants. 
Although it is the original definition of a compound determinant which 
is given on starting, the name afterwards seems to be unconsciously limited 
to compound determinants whose elements are minors of a given deter- 
minant of the n th order. This limitation leads to the introduction of the 
word class in connection with compound determinants to indicate “the 
degree of minority of the constituents ” (i.e. elements), a compound deter- 
minant of the i th class being one whose elements are minors of the (n—i) th 
order ; thus, the adjugate determinant is a compound determinant of the 
n th order and 1 st class. If a compound determinant of the i th class be of 
the highest possible order, namely, the 
n(n— 1) ... (n-i+ 1),^ 
1-2 ... i ’ 
— that is to say, contains all the minors of the (n — i) th order — Spottiswoode 
* And knowing this he might have indicated another mode of proving Cauchy’s extended 
multiplication-theorem. For example : 
1 a x a 
\ b x b 
2 1 . 1 X 1 
2 1 1 2 /l 
x 2 \ la 
2/2 1 + 
1 «3 
l ^3 
il 
X \ X 3 1 
2 /i 2 /s 1 
+ | 
a 2 a 3 \ . 
b 2 63 1 
CL 2 CL% 
• 
• 
K 
^2 ^3 
= 
X 1 
Vi 
1 
• 
x 2 
2/2 
1 . 
X 3 
2/3 
. 1 
= 
- a x x x - 
- a 2 x 2 — 
a 3 X 3 
- 
a \Vi 
— a 2V2 ~ ' 
« 3 2 /s 
-Vv 
- b 2 x 2 — 
boX% 
- 
\Vi 
— ^ 22/2 “ 
^ 32/3 
. . . 
x i 
2/l 
1 . . 
x 2 
2/2 
. 1 . 
X 3 
2/3 
. . 1 
| a l x 1 + a 2 x 2 + a 3 x 3 a^^ + a^+a^y^ 
I b x x x +b 2 x 2 +b s x 3 b l y l + b 2 y 2 + b 3 y 3 ■ 
