Dr Muir on General Determinants. 
689 
1907-8.] 
Taking n = 4t and r, s, p, a — 1 
writing them thus : — 
, 2, 
3, 4, we can best illustrate these by 
*21 a 33 ^44 I ’ ! ^11 ^22 ^33 ^44 I | ^12 ^22 ^32 ^4 
b 12 | r. 
22 C 33 C 44 I 
°21 °22 u 23 °24 
C 31 C 32 ^33 C 34 
I a c 
' ! ^11 b-22 ^33 ^44 
— I ^12 ^22 ^32 ^42 
^14 ^24 ^34 ^44 
'41 c 42 1 43 °44 
— | ^> 12 ^24 I ' I c : 
23 °44 
I + 
I u '21 ^33 ^44 I 
I ^11 ^23 ^44 I 
a 21 a 32 a 43 I 
^11 ^22 ^43 I 
^12 
^22 
b \2 
- 
*14 
*24 
*44 
C 21 
^22 
a 24 
C 24 
C 21 
^22 
$22 
C 24 
C 31 
C 32 
a 34 
C 34 
C 31 
C 32 
a 32 
C 34 
^42 
a 44 
C 44 
C 41 
^42 
(^42 
C 44 
- 1 
^24 
*i 2 | 
* 1 C 32 C *44 
1 + 
The right-hand member of (1) is equivalent to a direction to substitute for 
the v th row of R the s th column of Q ; similarly, the right-hand member of 
(2) is a direction, though not so evident, to substitute for the r th and p th rows 
of R the s th and a- th columns of Q ; and it is clear that (1) and (2) are but 
the first two identities of many. On the other hand, (3) is quite diverse in 
character, being got by the combination of two results analogous to (2). 
This is best brought out by noting that in the examples the right-hand 
members of (1) and (2) are got by multiplying 
1 
1 
®21 
CL 2 2 
^23 
a 24 
and 
a 21 
®22 
a 23 
a 24 
a 31 
a 32 
a 33 
a 34 
1 
«41 
<^42 
a 43 
«41 
a 42 
a 43 
«44 
respectively by | b n b 22 b 3S b u | ; and that similarly the two four-line deter- 
minants on the right of (3) are got by multiplying 
and 
1 
*n 
^12 
*13 
*14 
^21 
a 22 
a 23 
^24 
by 
*21 
^22 
^23 
*24 
a 31 
a 32 
a 33 
%4 
1 
«41 
a 42 
a 43 
^44 
*41 
^42 
h 43 
*44' 
1 
*n 
b\2 
b\3 
*14 
a 2\ 
®22 
a 23 
a 24 
by 
*21 
^22 
^23 
*24 
a 3\ 
a 32 
a 33 
a 34 
1 
«41 
a 42 
a 43 
«44 
*4! 
b\2 
^43 
*44 
VOL. XXVIII. 
44 
