213 
1907-8.] Primary Minors of an n-by-(w-hl) Array. 
(4) A direct proof of theorem (II.) is not easily formulated: in fact, 
determinants of the kind which it concerns, namely, those whose elements 
are products of determinants, seem to have been little studied. The 
following case, in which the factors of the said elements are the principal 
minors of a given determinant, is, therefore, of interest. 
The given determinant being | af> 2 c 3 d 4 \ , or A say, and its adjugate 
being denoted by | AjBgCy^ | , the determinant for investigation is 
A]A 2 
A 2 A 3 
A 3 A 4 
a 4 a, 
BiB 2 
^2^3 
b 3 b 4 
C 2 Cg 
C 3 C 4 
c 4 c, 
d,b 2 
^2-^3 
D 3 D 4 
d 4 d, 
Expanding it in terms of minors formed from the first two columns and 
minors formed from the last two columns, we obtain 
A 2 B 2 1 A 4 B 3 I • C 4 D 4 1 C 3 D 4 j - A 2 C 2 1 A 4 Cg I . B 4 D 4 1 BgD, | + A 2 D 2 1 A 4 D 3 1 • B 4 C 4 1 BgC, | 
+ B 2 C 2 1 B 4 Cg | • A 4 D 4 lAgD, | - B 2 D 2 1 BjDg | • A 4 C 4 1 AgC, | + C 2 D 2 1 C^Dg | • A.BJAgB , \. 
But as | A 4 Bg ] = — | c 2 d 4 | A , | C 4 Dg | = — | a 2 b 4 | A , .... this is trans- 
formable into 
r — A 2 B 2 1 c 2 d 4 | • C 4 D 4 | | + A 2 C 2 j b 2 d 4 | • B 4 D 4 1 a 2 c 4 | 
— B 2 C 2 1 a 2 d 4 j • A 4 D 4 | b 2 c 4 | + B 2 D 2 | $ 2 c 4 | • A 4 C 4 j b 2 d 4 | 
— A 2 D 2 1 b 2 c 4 | • B 4 C 4 1 ct 2 d 4 j 
— C 2 D 2 1 | • A 4 B 4 | & 2 c 4 | ; 
and as the expression here bracketed is the similar expansion of another 
four-line determinant, we have the interesting identity 
A 1 A 2 
A 2 A 3 
A 3 A 4 
a 4 a, 
cl 2 A 2 
C^4 A 2 
“A 
« 2 A 4 
BjB 2 
B 2 Bg 
B3B4 
b 4 b, 
= A 2 
&2 B 2 
^ 4 B 2 
\ b 4 
K B 4 
CA 
C 2 Cg 
C 3 C 4 
CA 
C 2^2 
c 4 C 2 
c 4 C 4 
r 2 C 4 
DiD 2 
AA 
D 3 D 4 
d 4 d 4 
d 2 D 2 
dfl 2 
where the determinant on the right is got from that on the left by putting 
a 2 , b 2 , c 2 , d 2 for A x , B 1 , C 4 , , and a 4 , b 4 , c 4 , d 4 for A 3 , B 3 , C 3 , D 3 . 
Taking the complementary of the identity (III.), we have 
a Y a 2 
a 2 a 3 
a 3 a 4 
a 4 a Y 
® 2 A 2 
« 2 A 4 
«A 
^ 4 A 2 
bf) 2 
\\ 
•A 2 = 
& 2 B 2 
&4®4 
& 4 B 2 
C 1 C 2 
c 2 c 3 
C 3 C 4 
c 4 Ci 
C 2^2 
r 2 C 4 
c 4 C 4 
c 4 C 2 
dft 2 
d 2 d 3 
d 3 d 4 
dfD 2 
c? 2 B 4 
d i l h 
d 4 B 2 
(IV.) 
