The theory of determinants in the historical order of development, by Sir Thomas Muir.

ALTERNANTS (NAEGELSBACH, 1871)

145

and therefore
= {C-+1(al,... an)+xCfh(al,... a)
= (x-a()(x-2)... (x-an).(al an)}
' (C_- h+1 + Z-Cn _ ) * ( Xnn- -1 + C2xn-2-. )2.
He then equates the coefficients of x1 in the two members of
this, obtaining. 1 - 1 7.11ti - i 1 -1 —i

a1. a- 1 aw1... a'- a
(_1)n-j+i a2 a2 a2 a2.. a a 2
0 1 h-)-{C l h+1j -1 j
nf n * n n ~ n 2 ^
( ) n-7L+1 CnAj - Cn-4 Cn-eiln

a1
5+1
a2
n+l
n

and thence
o 1 h-l +1 j -1 +1 e+1 C+_1 C n- nj
a1 a2... ah a....' a.+' an+' 2-' f
2. h+1 - -1 i n C+21_, C n-h
Following the same procedure with this second result as a basis,
he next obtains
| 0 1. h.. a -1 7h —1 j -.1 a k- 1 a k * n+2
C~n+2-h Cn+1-h C -_ _
and thereafter in similar fashion
0 1 h-1 h+1 — 1 + -1 1..+1 1- 1+1 n.13
a1 a. a a a a... a a a a a
O^h 1 2 ^ ' ' a k —2 ^k-1 * * * 1-3 -2 * n
Cn+3 -1 Cn+2 - zCn+l - C,1 -_ Cu+3- 72; C+2-fc kGn+l-k Cn-k
C C C
Cn+3-j C_+2-j Cn+1-j Cn-j
C,+3- +2 -h Cl+l- C_7 a-h
The general theorem which Naegelsbach thus arrives at * is
perhaps best enunciated in the form of a 'rule,' namely, The given
* Naegelsbach burdens himself with a troublesome, sign-factor which, by two
simple changes, we have rendered unnecessary.
M.D. III. K