83 
1906-7.] Dr Muir on Minors of a Product-Determinant. 
third order belonging to this class. Taking next the determinant which is 
the product of two determinants of the fourth order | a-p 2 c z d 4 [ , | j , 
we see that the two-line minor belonging, say, to the 3rd and 4th rows and 
to the 1st and 4th columns is 
C l C 2 C 3 C 4 
/i u 
ffl 94 
ll-y h 4 
ky k 4 , 
and that this is equal to 
1 C 1 dy 1 ; 1 C 1 ^3 I > 1 ^4 1 l 1 C 2 ^3 1 ’ | ^4 1 ? 1 C 3 djy ] 
l/l 94 I ’ l/l ^4 I ’ l/l K\> I 9\ ^4 I J I ^1 ^4 | 5 | k 4 I 
a result exactly similar to that of § 1 but not expressible in the same final 
form, on account of the two-line determinants occurring in it being no 
longer principal minors. Taking, in the third place, the determinant which 
is the product of three four-line determinants 
I «i \ C 3 d 4 I » l/l 92 K k i\> I m i n 2 r 3 S 4 i > 
we have for the two-line minor belonging, say, to the 1st and 3rd rows and 
to the 2nd and 3rd columns the expression 
a i a 2 a 4 
/l S 2 f% 9 4 
m 2 m 3 
Cy C 2 Cg C 4 
9i 92 9s 94 
n 2 n z 
hy h 2 h z h 4 
r 2 T 3 
ky k 2 k 3 ky 
S 2 S 3 
which is equal to 
ay a 2 
a 3 a 4 
a-, a 2 a 3 a 4 
Cb-y Ct^ ^3 ^4 
ay a 2 a 3 a 4 
m 2 m 3 
n 2 n s 
fi 9i 
/o .-/ 2 k 2 k 2 
/3 9z k 3 k 3 
/ 4 9 4 K K 
Cy C 2 
c 8 c 4 
<M 
^1 ^2 ^3 ^4 
Cy C 2 Cg C 4 
r 2 r 3 
fl ."l 
hy ky 
/ 2 92 k 2 k 2 
/s 93 h 3 h 
/ 4 94 h 4 K 
S 2 S 3 
/l/ 2 
* 1 m 2 n z | + 1 a 4 a 2 a 3 a 4 
/l / 3 
9i 92 
1 C 1 c 2 C 3 C 4 
ft 93 
hy h 2 
liy h. 
ky k 2 
ky k 3 
+ 
ay a 2 a 3 a 4 
Cy c 2 c 3 c 4 
/ 3 f 4 
9z9 4 
7ig h± 
k 3 k 4 
' K S 3 
