302 
Proceedings of Boyal Society of Edinburgh. [june 20 
This convenience, however, is a rare one ; for the case when 
n = - J, that is when the generating condition is 
\0z — — ^.i z 6z z. % z 0z 
the expression becomes 
„ A f I Z 
0z — A. I 1 + vr ~ 
3z 2 
f 
5z 3 
7z 4 
1. 2.3.4 1....6 1 
3 + &C. } 
for which no such facility presents itself. 
For the purposes of calculation this may he written 
i+l-(>Th+()A-oA+&°- 
z 
0 
z 
376 
z 
5T8 
where the mark ( ) stands for the precedin 
term. 
z 
6z 
+ 
8 
— 
•080 
0038 
+ 
7 
+ 
•379 
2051 
+ 
6 
+ 
•773 
2548 
4- 
5 
+ 
1-141 
9518 
+ 
4 
+ 
1-402 
4478 
+ 
3 
+ 
1-549 
0238 
+ 
2 
+ 
1-552 
8556 
+ 
1 
+ 
1-381 
7733 
0 
+ 
1-000 
0000 
— 
1 
+ 
•367 
8795 
— 
2 
— 
•558 
4152 
— 
3 
— 
1-827 
1822 
The first or lowest intersection is between z— -2 and z= - 1, 
the second between z = + 7 and z — + 8 ; but the position of the 
third has not been examined. 
r 1 
When n is a rational fraction such as - , if we make a = -j, the 
s s 
function takes the form 
, 1 Z 11 
<t>Z— 1 h 
1 1 
Z‘ 
r 
r r + s Is. 2s r r + s r + 2s Is. 2s. 3s 
&c. 
Z 
=i_()l —+o — 
r s r + s 2s 
-() 
1 
z 
r+2s 3 s 
+ &c. 
