440 Proceedings of Royal Society of Edinburgh. 
the general term being 
[sess. 
F(fa][/J]M) • F(HM[y + 1]) .... F([«][ffl[y + r]) • 
(2.) 
The following notation will be employed : 
w 
denotes 
P n ~ 1 
V -1 
[*]• 
p - \ ' p 1 - 1 ’ p 3 - 1 . . 
. . p 1 — 1 
5) 
( p-lf 
In 1 
P n — 1 ■ p"- 1 —1 ■ pn-2 - 
1 . . . p n -r+l- 1 . 
l r J 
5? 
P ~ 1 ‘ P 1 — 1 ’ P 3 — 
1 ... p r - 1 . 
{ a l 
„ L 
K = CO 
pa-n+l _ ^ • pa- n-\- 2 _ ^ 
pa — n-\-K ^ ’ p n ~ FI --- ^ 
l » J 
p a -\- 1 — ^ • pa+2 _ ^ 
Ps 
r— 1 
1 
+ 
e 
• 1 rv\R'~TK. 1 
P 1 .... p 1 v n(a-n) 
■ p* - i p* - 1 1 
It is assumed that p x when x is not integral means its absolute- 
value. The four infinite products in <j ^ | are each of them con- 
vergent if p is greater than one. Thus | ^ j> is finite single valued,, 
etc., unless a is a negative integer. In the exceptional cases ( a l 
is infinite or indeterminate. When n is a positive integer 
reduces to 
pa ][ ' p a ~ 1 1 • p a ~^ 1 pa — n -\- 1 ^ 
p — 1 ‘p 2 - 1 p 3 - 1 .... p n - 1 
/ a ( 
\ n f 
DW denotes the operator 
d 
d(xr r ~ 1 ) I d(xP r -*) ' ' 
d f mm I 
d{x&) I d(xP) (dxl f / ■ 
( 3 > 
