372 Proceedings of Royal Society of Edinburgh. [sess. 
The Sum of the Signed Primary Minors of a 
Determinant. By Thomas Muir, LL.D. 
(MS. received July 25, 1904. Read November 7, 1904.) 
(1) The fundamental propositions in regard to the sum of the 
signed primary minors of a determinant are — 
( A) An expression for the negative sum of the signed primary 
minors of any determinant is got by taking a determinant of the 
next higher order whose first element is zero with the given deter- 
minant for complementary minor , and whose remaining elements 
are units all positive or all negative. 
(B) The sum of the signed primary minors of any determinant is ex- 
pressible as a determinant of the next lower order , any element (r , s) 
of the latter being the sum of the signed elements of a two-line minor of 
the former , viz., the sum (r, s) - (r, s + 1) - (r + 1 , s) + (r + 1 , s + 1 ) . 
(C) If the elements of a determinant be all increased by the same 
quantity w, the determinant is thereby increased by w times the 
sum of its signed primary minors .* 
(2) By the application of the first of these the following results 
are readily obtained — 
The sum of the signed primary minors of the alternant 
| a°b a cP . ... \ is equal to the alternant itself. (I) 
The sum of the signed primary minors of a circulant of the \\ th 
order is equal to n times the quotient of the circulant by the sum of 
its variables. (II) 
Thus, the sum of the signed primary minors of C {a , b , c) 
1 
1 
1 
— - 
1 
1 
1 
1 
a 
b 
c 
a + b + c 
a 
b 
c 
1 
c 
a 
b 
a + b + c 
c 
a 
b 
1 
b 
c 
a 
a + b + c 
b 
c 
a 
-3 1 
a 
+(a + b + c) , 
#;:($ + b + c) , 
1 1 
b c 
a b 
c a 
= 3C(a, b, c) f, ( a + b + c ). 
* Proceedings Roy. Soc. Edinburgh , xxiv. pp. 387-392. 
