198 
Proceedings of Royal Society of Edinburgh, [aprtl 18, 
a l 
a 2 • • • 
a 
b 2 
h ••• 
b n +i 
C 2 
Co . . . 
o 
C n+ 1 
or 
~ i a i 
b‘ 2 C 4 • • • l n + i 1 
h 
l 3 ... 
ln+1 
a i 
a 2 . . . 
«« 
h 
b. 2 ... 
bn 
C 2 
• 
c 3 . . . 
• 
^n+1 
or X 2 
= K 
c 4 • • • ^w +1 I 
l 2 
h ••• 
ln-f - 1 
a ! 
a 2 * * * 
O'n 
h 
b 2 . . . 
K 
c i 
C 2 ' . ' 
C n 
d 2 
rl 
• • • 
d’n+1 
or S 3 
= | a x 
b-2 ^3 dfr . . . l n+1 
h 
X ■ ■ • 
l’n + 1 
• 
• • • 
«i 
• • • 
«» 
b 2 . . . 
b n 
• 
* 
* 
or X n . 
-i = 1 
b% Cg . . . ll n _ 2 l' n -|- 
/( 1 lie, ... 
L l 
3 
ln-\ 1 
where, generally, ^ consists of determinants in each of which 
the suffix-order (1 2 3... n) occurs /x times, — „CV denoting the 
number of combinations of n things taken g together. 
We now propose, with the help of these identities, to effect the 
summation of certain series of Alternants, obtaining results which 
may be called Extensions of the known theorem referred to at the 
outset. 
