536 Proceedings of Royal Society of Edinburgh. [sess. 
(4) An especially interesting case is that where r—n- 1, the 
theorem then being — The sum of the primary minors of A 2 is equal 
to the sum of n squares , each of which is the square of the sum of 
the elements of a column of the adjugate of A. This may he 
established independently by starting from a known theorem 
regarding the ‘bordering’ of the product of two determinants 
(Messenger of Math., xi., year 1882, pp. 161-165). For the 
third order this theorem is 
a f\ c 2 1 ' I a o/h72 1 4" I 1 1 a ofiiYs ! 
+ I a o^2 G s I ■ I “Ays I > 
and making an evident specialisation we have 
1 1 1 
1 
«1 
a 2 
2 
1 
«1 
a 3 
2 
1 
a 2 
a 3 
«lVs I 2 
= 
1 
K 
^2 
+ 
1 
h 
+ 
1 
b 2 
h 
1 
c i 
C 2 
1 
C 1 
C 3 
1 
C 2 
C 8 
Now the left-hand member here is equal to the sum of the signed 
primary minors of | a 1 b 2 c 3 1 2 ( Proc . Roy. Soc. Edin ., xxiv. pp. 
387-392 ) ; and the right-hand member is equal to 
(a, + b 3 + c 3 y + (a 2 + b 2 + c 2 y + (a, + b, + c,)*. 
That 4 unsigned ’ may legitimately be substituted for 1 signed ’ is 
made evident on bordering with 1 , - 1 , 1 instead of l , 1 , 1. 
(5) In the theorem of § 3 it is the sum of all the r-line minors 
that we are concerned with : there is, however, an equally 
important theorem when we confine ourselves to the coaxial 
minors. It is — The sum of the coaxial r-line minors of A 2 is equal 
to the sum of the squares of all the r-line minors of A. 
No formal proof need he given in view of what has come to 
light in proving the other theorem. Merely as an illustration we 
may note that when A = | afb 2 c 3 d A \ and r — 2 we have the sum of 
the two-line coaxial minors, 
11 121+ 
11 13 
+ 
11 14 + 
22 23 j + 
22 24 
+ 
33 34 
12 22 
13 33 
14 44 
23 33! 
24 44 
34 44 
Ro Yo 
I <*i Vs 
\ I a l/^2V3 I 
