1904-5.] Signed Primary Minors of a Determinant. 
381 
It follows, therefore, because of the known result 
that 
k fl =(-r + v 
/ _ \n+l^n 
1 a” +1 | 
i jS»+i ! 
a — (3 
a n+ 1 - /5 n+1 
i i 
1 a 
n + 1 
1 + a + . . . + a! 
(XVIII) 
1 /5" +1 1 +/5 + . . . + /5" 
This curious result ought to agree with (XVII) : in other words, 
we ought to be able to show that 
( _ 
a-7* 
11 n + 1 
1 a n+1 1 + a + . . . +a n 
1 /5 W+1 1+/5 + . . . +j3* 
wK w + (6 + c)K n _! 
-{6 2 + c 2 )K„_ 
Towards doing so it has first to be noted that the determinant on 
the left 
1 1 n 
1 <x n+1 a + a 2 + . . . +a n 
1 j8 n+1 /i + /5 2 + 
1 a w+1 | + 
1 
1 
1 /3 n+1 
1 
a" +1 
a + a 2 + . . 
. • + a” 
1 
f3 n+l 
/3 + /3 2 + . . 
, . +/3” 
//, 
1 a" +1 
+ 
a” +1 a + a 2 + . 
,. + a n 
- 
1 a H- a 2 + . . 
• + a* 1 
1 /3 n+l 
/8“ +1 /3 + fS 2 + . . 
.+/?• 
1 /3 + /3* + . . 
•+H 
= 71 
1 a w+1 
1 /5 W+1 
+ . 
7i I 1 a n+1 
| 1 f3 n+1 
1 ! /3 n+1 A’ 
| /3 n+1 
-11 a w 
- 
1 a n ~ x 
i 1 
1 /5”- 1 
- (a/5 + 1) 1 1 
a" 
— (a 2 /? 2 +1)11 a”- 1 
i p* 
! 1 /5 ?t-1 
= - n 
( a *+i - ^”+1) + (^+ l)( a » - /3”) + (^ + l)(a tt_1 - ^”- 1 ) + . . . . 
Multiplying this now by (- ) n+1 c n /(a - f3) and substituting K n for 
( - e) n (a n+x - /5 n+1 )/( a - /l) we obtain 
