106 Proceedings of Royal Society of Edinburgh. [sess. 
Using it n - p + 1 times in succession we have 
II 
CO 
a, 
i 
52? 
+ 
M P| , , 
— -p-i . A s +h — 
- M p +2,S+2A 
- 
Nn_^_2 . As+2h = 
+ 
^p+2,s+2A 
— • ^s+3/i = 
- 
^lp+3,s+3^ 
( y i . A-s+(n-p)h — 0 + ( — ) /l n,s+(n-p)h 
and therefore by addition 
or 
I . a+h . a+2h 
! A i K 
h . h 
h . h 
h . h 
n-p-1 • As+h + ^n-p 
-2 • As + 2h ~ • ■ 
1 
4 
0 
0 . 
a+{p - 1 )h a+(p+l)h 
. a+nh . 
... A , A 
s ) 
>-l A i> 
n-1 
ns 
Jl's+h-P) 
\\ a+h 
a+(n-l)h s , 
* ■ ‘ * ^n) 
1 1 ' ' ‘ * 
■"n-1 J 
^hyn-p-1) j 
a+h 
! A i • • • • 
a+(n-l)h s+h x 
A n-1 A n J 
h\(n-p-2) r 
’ ’ * ' x ns 
|a“ +, ‘ . . . . 
. a+(n — 1 )h . s+2/i\ 
A «-i A n ) 
1 -,x n ~P( X h a h i \ a + h a+(n-l)h s+(n-p)h^ } 
+ \ A ' V A 1’ A 2’ • ' • * A n/ * 1.1 A n . J 
a theorem which may be described as giving an expression for 
an alternant having two breaks in its series of indices in terms 'of 
alternants which have only one such break and that at the very 
last index. On account of the fact, however, that alternants of 
the latter kind are multiples of the alternant which has no break 
at all — that is to say, on account of the theorem 
[A' 
h h 
1’ A 2’ 
1(P) 
a+h a a+2h 
=!|Aj + '‘a“ +2;i 
. a+w/i\ 
• K ) 
. a+(n-l)h a+(n+^\ 
• * n - 1 A n ) 
above given as an important special case of the general theorem of 
the third chapter — substitutions may be made which will result in 
the appearance of the last mentioned simple alternant in every 
term. Consequently, if we divide by this alternant and [put 
s = a + (n + m)h we have the theorem 
