1902-3.] Dr Muir on Generating Functions of Determinants. 391 
determinant for supplementary minor, and whose remaining ele- 
ments are units , all positive or all negative. (IX) 
(b) The sum of the signed primary minors of any determinant 
is expressible 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 + l,s) + 
(r + 1, s + 1) ; and, therefore , if the former determinant be axisym- 
metric , so also is the latter. (X) 
(c) If the r th row of any determinant A of the n th order be 
replaced by a roio of units and the resulting determinant be denoted 
by D r , then the sum of the signed primary minors of A is 
(so 
r=l 
(d) If the elements of any roio of a determinant be all alike , 
the sum of the signed primary minors reduces to that of the minors 
which are complementary of the elements of this row (XII) 
(e) If the elements of any row of a determinant be all increased 
by the same quantity w, the determinant is thereby increased by 
cd times the sum of its signed primary minors. (XIII) 
(f) The sum of the signed primary minors of the determinant 
which is the negative sum of the signed primary minors of any 
determinant A is equal to A together with twice the sum of the 
signed primary minors of A. (XIV) 
The first of these theorems, (a), is proved by using the develop- 
ment whose typical term is the product of an element from a 
fixed row, an element from a fixed column and the secondary 
minor connected with these : the reduction of order effected in 
(b) is due to the existence of a row of equal elements and a 
column of equal elements in the determinant of (a) ; the truth 
of (c) is shown by expanding each of the determinants under 
2 in terms of the unit-elements and their complementary minors ; 
( d ) follows as a corollary from ( c ) ; (e) is dependent on the same 
theorem as (c) and on (c) itself ; and (/) is a consequence of (a) 
and the case of (e) where w is 1. 
The application of (b) to the coefficients in (VIII) gives 
1 - acx 2 
ax)( 1 +cx)(l - bx + acx 1 ) 1 
1 
+ (2b - a-c)x + 
12 b-a — c b — 2a\ 
7 o oz. r'" + 
j b - 2c 2 b - a - c | 
