1905 - 6 .] The Y-line Minors of the Square of a Determinant. 537 
a l 
a 2 
«8 
a 4 
2 + \a 1 
a 2 a z 
a i p+ 
\ 
b 3 
h 
1 tq 
C 2 G z 
C J 
= | af 2 12 + 1 afi 3 p + | af, p + 1 a 2 b 3 p + | a 2 b 4 P + 1 a z b 4 p ' 
+ 1 a^ 2 P + 1 ttjCg P 4- 
v 
+ 
+ \<hd 2 P + kirf 8 P+ 
as the theorem specifies. 
(6) The theorems of §§ 3, 5 may be further widened by passing 
from A 2 to AjAg, the results being 
The sum of the Y-line minors of A : A 2 is equal to the sum of j 
products , each of which has for its first factor the sum of the Y-line 
minors formable from r columns of A x and for its second the sum 
of the corresponding minors of A 2 . 
r The sum of the Y-line coaxial minors of AjA 2 is the sum of all 
possible products , having for their first factor an Y-line minor of 
A 1 and for their second the corresponding minor of A 2 . 
Thus if A 1 ee | af 2 c 3 | and A 2 = j cq/^yg | , the sum of the two- 
line minors of AjA 2 is 
{ I I d* I a l C 2 I d* | | } * { | a i^2 I d" | a x y 2 | + | f3 l y 2 | } 
+ { l%l + l“i c sl + I Vsl} ■ { l“AI + I“i7sl + I Aral } 
+ { I a 2^3 I + I a l c 3 I + I V ; 3 I } • { I a iPs \ + I “273 I + I Pi/3 I } > 
and the sum of the two-line coaxial minors is 
I a i^2 I ' I a i^2 I d- | a 1 b s | • | a^g j + j a 2 b 3 | • j a 2 / 3 3 | 
d* | a i c 2 I " I a l72 I d- I a Y C g | • | cqyg | + j a 2 Cg I • I a 2 y 3 j 
d“ I ^\ c 2 I * I PiY 2 I d- | fiiC 3 | • | f3 1 y 3 1 + | b 2 c 3 j • | fS 2 y 3 1 . 
(7) The ultra-symbolical expressions used in §§1,3 suggest that 
a freer use of non-quadrate arrays might be advantageous. We 
might, for example, use them as elements of a determinant, 
thereby arriving at such identities as 
= KVel + NaVeU 
(«1 , <h) («3 > a i ) ( a S » a ,i) 
Qi,h) Os > h) (h,h) 
(Cj , C 2 ) (Cg , C 4 ) (C 5 , Cq) 
