1905-6.] The r-line Minors of the Square of a Determinant. 533 
The Sum of the ?*-line Minors of the Square of a 
Determinant. By Thomas Muir, LL.D. 
(MS. received September 3, 1906. Read November 5, 1906.) 
(1) If we temporarily denote the product of the r th and rows 
of a determinant A by rs, we shall have conveniently 
°1 
a 2 
a s 
« 4 
2 
11 
12 
13 
14 
*4 
h 
h 
&4 
12 
22 
23 
24 
<V 
C 2 
C 3 
C 4 
13 
23 
33 
34 
*1 
d 2 
d 3 
14 
24 
34 
44 
The sum of the elements of the latter determinant is evidently 
11 + 22 + 33 + 44 + 2.(12 + 13 + 14 + 23 + 24 + 34), 
and therefore may be written in ultra-symbolical form as a square, 
namely 
(1 + 2 + 3 + 4) 2 , 
or, with greater and quite sufficient fulness, 
{ (®i j ^2 » ^3 > ^ 4 ) "b (^1 > ^2 ’ ^3 » ^ 4 ) *b (^1 j ^2 ’ ^3 j ^ 4 ) “b (fi 5 ^2 ’ ^3 ’ ^ 4 ) 
This being equal to 
2( fl l > a 2> a 3’ a A a i’ a 2> a 3’ a ±) + 2 2( a i’ tt 2> «3> «4 iS & 1 > b 2 1 & 3> & 4 )> 
let us attend to the terms in it which only contain letters with 
the suffix 1. Of these there are manifestly under the first 2 
n 2 h 2 r 2 (12 
5 ) Uj \ 5 
and under the second S 
2 afj-L , 2 a x c, x , 2« 1 c? 1 , 2^^ , 2 5 1 c? 1 , 2^^ : 
so that the aggregate is (a 1 + b 1 + c x + df 2 . Similarly the sum of 
the terms in which only the suffix 2 occurs is ( a 2 + b 2 + c 2 + d 2 ) 2 ; 
and so as to the other suffixes. Further, there are no terms 
involving a variety of suffixes : consequently we have as the full 
result 
(cq + \ + c x + f) 2 + (a 2 + b 2 + c 2 + d 2 ) 2 + (a 3 + b 3 + c 3 + d 3 ) 2 
+ (a 4 + & 4 + c 4 + d 4 ) 2 ; 
