1904 - 5 .] Signed Primary Minors of a Determinant. 
379 
(9) If it be desired to have the general result arranged 
according to descending powers of a, we have only got to sub- 
stitute in Xn-\ — (p + c)xn ~2 + (& 2 + c2 )x «— 3 ~ • • • . the expressions 
for x»-i , Xn -2 j • • • obtained from (XIY), and then collect the 
coefficients of like powers of a. The theorem thus arrived at is — 
The sum of the signed, primary minors of 
b b 
a a a . 
c e 
is 
na n ~ 1 
- (n- \)a n ~ 2 {b + cj 
+ (n- 2 )a w-3 {(6 2 + c 2 ) - C n _ x , fbc) 
-(n- 3)a n ~ i {(b 3 + c 3 ) - C n _ 2 , fb + c)bc\ 
+ {n - 4K~ 5 {(6 4 + c 4 ) - C w _ 3 , fb 2 + c^bc + C„_ 2 , 2 b 2 c 2 } 
- (n - 5 )a n ~ G {{b° + c 5 ) - C w _ 4 , x (6 3 + c 3 )6c + C w _ 3 , 2 (6 + c)6 2 c 2 } 
+ (XV) 
The cofactors here of na r 1-1 , —{n— 1 )« n_2 , . . . are related to 
one another in a curious way, which is worth noting if only for 
use as a check in computation. Denoting them by X , X x , X 2 , . . . 
we have 
X^m+l = (P T m | 
X 2m = ( J + C)X-ton-l - ( re + 1 ) — m_i • b m C m j < XV1) 
the demonstration of both resting on the facts 
(b r + c r )(b + c) = (b r+1 + c r+1 ) + bc(b r - 1 + c r ~ 1 ), 
Qp , q ~ ^p—1 , q T C Jp—\ ( q—\ . 
(10) It is thus suggested to examine the result of multiplying 
the whole expression by a + (b + c). Taking it in its original 
form 
Xn-1 - (b + c)xn- 2 + {b 2 + C 2 )Xn -3 ~ • • • • 
we readily see, to begin with, that the product is 
a Xn - 1 - a(b + c) Xn -2 + «(6 2 + C 2 )Xn— 3 
+ ( b + C )Xn- 1“ (b 2 + C 2 ) ) + (& 3 + C 3 ) 1 
- bc(b° + c°) j 2 + bc(b + c) j Xn “ 3 ‘ ‘ ‘ ' 
