1907-8.] Primary Minors of an ?i-by-(n+l) Array. 215 
and, therefore, by Laplace’s expansion theorem, 
1ST — — / — b 3 j AjBg | | a 3 c Y | | a 3 d Y \ | CjD 3 1 + c 3 1 A 1 G 3 | | a 3 b 1 | | a 3 d 4 | | B 4 D 3 | 
a 3 I ) 
— d 3 \ AjDgj | a 3 b 4 | | a 3 c 4 | | B^g j | ^ 
A 2 [ 
= — ) — ^3 | c 2^4 I I ffi C 3 I I | j <^2^4 I "t c 3 I ^2^4 I I I I ffi^3 | | a 2 C 4 I 
a 3 ' | 
— d | & 2 c 4 1 | aj&g | | a ]C 3 1 | a 2 o? 4 1 I . (VII.) 
Similar expressions are got by the like treatment of M, or simply by 
interchange of suffixes in (VII.); for example, by interchanging 1 and 3. 
Again, by performing on | a ± a 2 b 2 b 3 c s c 4 d 4 d x \ the operations 
we obtain 
a 3 col 4 — a Y col 3 , a 2 col 3 — a 4 col 2 , a Y col 2 - a 3 col x , 
b 2 | CL-fiz | b 3 | <^2^4 I ^4 I tt 3^1 I — 
C 2 I a i C 3 I C 3 I a 2 C 4 I C 4 I a 3 C l I 
d 4 | a 3 d 1 | 
d 2 a Y d 3 
d 3 | a 2 d 4 
which, by expansion in terms of the elements of the second column and 
their co-factors, gives us 
ill 
! a i a 2 ^2^3 C 3 C 4 = ) +^31^2^4! I ffi^3 1 I C 2^4 1 “ C 3 1 ^2 C 4 1 | ^2^4 1 
a 3 { | 
+ d 3 \a 2 d 4 \ 1®!%! | ^2 C 4 1 j • (VIII.) 
By the combination of those two results we are led to 
N = - A 2 • | ^2^3 C 3 C 4 d 4 d Y | , 
in agreement with (IV.) and (VI.). 
(7) There is another determinant which may be treated in the same 
fashion as | A 4 A 2 
B 2 B 3 C 3 C 4 D 4 D x 
i in § 4, 
namely, 
-A-2^3^-4 
A 4 A 3 A 4 
A 4 A 2 A 4 
AiA 2 A 3 
b 2 b 3 b 4 
BiB 3 B 4 
b 4 b 2 b 4 
BiB 2 B 3 
c 2 c 3 c 4 
CjCgC, 
CxCA 
CxC.Cg 
d 2 d 3 d 4 
d,d 3 d 4 
DiD 2 D 4 
DAD, 
The first result is 
!A 2 A 3 A 4 B 3 B 4 Bj C 4 CjC 2 DjD.D, i = A 2 
a 2^3^4 
aiA 3 A 4 A 1 A 2 « 3 
AiA 2 % 
b 2 B 3 B 4 
^B 8 B 4 B x B 2 6 4 
BiB 2 6 3 
C 2^3^ , 4 
C1C3C4 C 1 C 2 c 4 
Q\^2 C Z 
d 2 D 3 D 4 ^D 3 D 4 D ,D 2 d 4 DjDgDg ; 
(IX.) 
