1904 - 5 .] Signed Primary Minors of a Determinant. 
377 
— a result to be expected, since generally 
1 
1 
1 . . . . 
= 
«2 
a 3 • 
. 1 
1 
a 2 
« 3 . . . . 
\ 
h • 
. 1 
1 
b 2 ... . 
c i 
C 2 
e B . 
. 1 
1 
c i 
C 2 
c 3 ... . 
1 
1 
1 . 
= 
1 
I 
1 
1 
C 3 
C 2 
c i 
1 
h 
\ 
3refore 
1 
a z 
a 2 
a i 
M{ | af 2 c B ■•••!} — | • . • • of) 2 a^ | }. (IX) 
(8) When each of the a’s is equal to a, each of the b ’ s to b, and 
each of the c’s to c, the H’s and K’s are no longer distinguishable, 
and the expression (VI) becomes 
K n _ 1 + K n _ a .(l, b + ctK^-l) 
+ K n _ 3 • (1 , b + c, b 2 + c 2 jj K 2 , - K x , 1) 
+ 
+ (1, b + c, b 2 + c 2 , . . . $ K n _ 1 , - K n _ 2 , . . . ). 
This, however, is best arranged in portions containing 1 , He, 
b 2 + c 2 , . . . . , and their respective cofactors, the result then being 
(K n _ 
l? X M _ 2 , K n _g, . , 
. . , k 15 i § n , . 
• • » X M _g, K„_ 2 , K n _j) 
+ (b + c) ( 
X w _2) . . 
..,K v l$\,K v . 
• • 5 K n _ 2 ) 
+ (b 2 + c 2 )( 
4* 
151. Kj... 
■ • > X n _g) 
+ ... . 
(X) 
or, say, 
Xn - 1 
Now since 
- (S + C)X «- 2 
X" ^-n ^ ' 
+ (& 2 + C 2 ) Xn _ 3 
-iKj + K,_ 2 K 2 + 
and the known ultimate form of K w is an expression consisting of 
terms descending by second powers of a and ascending by first 
powers of be , viz. 
a n - C n _ ia a n ~ 2 bc + C M _ 2<2 a n_4 6 2 c 2 - , 
it follows that there must he for Xn an expression of similar 
