1907-8.] Primary Minors of an n-by-(n-\- 1) Array. 211 
is then 
. . d^d 2 
a-J) 2 + ajb-L 
(^ 1^2 ^2^1 * 1 
, . c 1 d 2 + c 2 d 1 
a i a s • ■ 
. . d^d 8 
+ a z h \ 
ftjCg + ttgCj . . 
. . c Y d 8 + c 3 c?-, 
a 4 a 5 . . 
. . d^d b 
a A + <X 5 6 4 
^ 4^5 "t" ^5^4 * * 
. . c A d b -I- c b d^ 
By performing on this the set of operations 
col 5 - ^J-coL - ^-col 2 , 
5 « i ^i 
col 6 - 'Xcoli - ^-col s , 
a i 
coL -i C ol.-^eol 4 , 
a 2 L d 1 
col 8 -fLcol 2 -^col 3 , 
h l 2 ®1 
col 9 - icol 2 - i-col 4 , 
“1 
CO, 10-7- Col 3-f- COl 4 ’ 
C 1 U 1 
we reduce to zero all the binomial elements in which 1 appears as a suffix, 
and replace every other binomial element by a fraction of the product 
of two determinants of the 2nd order, the reason being that 
(hk + mn)-±hm~ y -nk = (**-«*> 
y x xy 
= 0, when x,y = n,h or k,m. 
The determinant can thus be resolved into two factors, the first of 
which is 
«!« 2 
C 1 C 2 
d x d 3 
a l<H 
*A 
C l C 3 
C 0 
or 
a 4 a 4 
\ h 4 
C 1 C 4 
djd^ 
a i a b 
C \ C b 
d\d b 
and the second 
1 
KVA) 3 
1 ! ' 
■ 1 «A i 
i <Vi i ■ 
• | agCj | • • 
■ * 1 C 2^1 1 
• 1 1 
1 «2 & 1 1 ‘ 
■ 1 “A 1 
1 <h c l 1 
’ 1 tt 4 C l 1 * * 
• • | c 2 d x | 
• | c 4 c?, 1 
1 «A 1 ■ 
• 1 «A ! 
1 «4 C 1 1 ' 
■ 1 «5 C 1 1 • • 
• • 1 <A 1 
• | c b d Y | 
By establishing the fact that | ajb 2 c^d 5 \ is a factor of the original 
determinant, the former of these is the all-important : for we see that, if 
