223 
1917-18.] Determinants of a 4-by-8 Array. 
When it is pairs of columns that are interchanged the results are 
Cl 4" C2 4” ^13 4" 4" c 12 4~ ^13 = 
Cl “t ^22 4“ ^23 4~ C 21 + C 22 4" C 23 = 
Cl 4“ C2 + Cs C 31 4" C 32 4“ C 33 = 
a 
11 
= & X1 + b 2l + b 3l + c u + c. 21 + c 31 
= C2 4~ ^22 ^32 4" C 12 "t C 22 4" C 32 
— /> 13 + /) 2g 4- & 33 + C 13 + C 23 + C 33 , 
where there is the same simplicity, any line of | & 13 | with the corresponding 
line of | c 13 | taking the place of any line of | d u | in the former statement. 
13. Closely connected with these two sets are the following : 
Cl "t C 2 4~ ^13 = Al + Al ’ 
Cl + C 2 + C 3 = Al Al 5 
Cl 4" C 2 4~ C 3 — Al 4~ Al ’ 
C 11 
C 21 
C 31 
+ C 12 
+ C 22 
+ C 32 
+ c 13 = d n + Ai , 
+ C 23 = Ai 4~ Ai 5 
4~ C 33 = Al 4~ Al > 
and other six derived therefrom by changing each element into its 
conjugate. By addition we evidently obtain 
Ci 4~ C2 ^ C3 c n 4- r i2 c i3 = Ai 4~ Ai 4~ Ai + Ai — a u 5 
and the five others like it in § 12. 
14. From §§12, 13 we have simple expressions for the sums of the 
elements of the determinants, namely, 
2C — 2< hi 2 Ai > 
c = a 1( + > 
2A = Hi ; 
so that the sum of the 35 products is 8a n . 
15. A set giving a choice of expressions for the difference between one of 
the b ’ s and the corresponding c is still more interesting, as in the long 
notation the terms of the equalities seem quite devoid of order and are 
practically unrememberable. In one of its forms the set is 
11 - c n = a n 
21 ~ c 2i a w 
31 ~ c 3i = a n 
i 
A3 
+ 
d 
l 
+ 
A3 
+ 
A 4 
f 
+ 
A4 
l 
+ 
As 
+ 
A4 
/ 
A3 
+ 
d 24 
l 
+ 
A3 
+ 
A4 
'12 
F 
22 
C 12 “ a il 
- c 22 = a n 
f d 32 + d M 
l + A2 + d 44 
_ / A2 4~ A4 
l 4- A 2 4“ A4 
C2 c 32 — a n 
{ + d 32 + d 
“22 4" A4 
y 34 
]t _ _ n _ f A 2 + As 
°13 c i3“ a n 1 + d*n + d 
43 
'23 
C 23 — a il 
A2 "t As 
+ A2 ”1“ d, 
43 
^33 c - 
33 
= a 
f d^p + d, 
^ f + A2 4“ d 
23 
v 33 
where the square arrays stand for quadrinomials and are arrays of two-line 
minors of | A4 I • Each array is replaceable by another of the same 
kind, by reason of the fact that on account of the sum of the elements 
of any line of \ d u \ being constant , the sum of the elements of any 
two-line minor is equal to the sum of the elements of the complementary 
minor. 
