3 -f 4 
i “f” ai 
322 Mr. Robertson's new Demonstration 
Proceeding therefore as in the 17th and 18th articles, and 
I 
# n .. n I tr-rrr f J J 
using the same notation, 1 — x\ =1— — #+ ~ • 
— ■ ■ x 2 — ~ 
z r 
1 1 1 2 
1 1 1 
x z 3 
, , 1 r r r . 0 
X A . . . *2’ — &c. 
* r 2 
3 4 
n 
Also 1-— xi r — 1— “ AT 4 - ~ . X'— 
T * Y 2 
n n 
nr r 
X 1 
« « « 
■ 1 2 3 
.nr r r 4 
“I™ — , — — — — — — — — — ^ oCC. 
1 r 2 
3 4 
20. It is easily proved, by means of the 15th and 16th 
articles, that 
1 -j- mx~\~m . -j— x*A r m . — . — — x 3 -l~m . — — . — y- . — x -f- &c. 
or- — — — 
1 -f- nx-\- 71 
n — 1 
2 
n — 1 n — 2 
X 3 -\-7l 
n — 1 n — 2 n — 3 
is equal to the series 1 — n. 
2 3 
m — n — 1 
a' 4 -f- &C. 
X 2 -\ -7U 71 
m—n — 1 m — n — 2 
x -\- 7 n — n . 
m — n — 1 m — n — 2 m — n — 3 4 
a’ 4 -|- &C 
3 k 2 3 4 
whether m and n be whole numbers or fractions. For v being 
equal to m—71, this last series becomes F-j-z; . 
V— l V— 2 
2 
j? 3 -|-z> . v —^~ . v —^ . a' 4 -f- &c. ; and this series being 
multiplied by 1 ^nxA^n . • -7- • -y- •»*+» . n —^~ • yy 
2 » 2 3 ■ 2 
See. the series expressing their product, by the 15th 
v + n — 1 
and 16th articles, is . 
.x*j^.vAj~n . 
v + n — 1 
V -\-n—\ 
X * 
v -f- n — - 1 w + m— ■ 2 — 3 
a ,4 -j- &c. But as 
3 2 3 4 
v is equal to m—n, this last series is equal to 1 . 
m — 1 , , m— 1 m — 2 , , m — 1 m — 2 m — 3 . 
X 4 ~ 771 . . x -\-m . - — — m 
3 
a ,4 -{- &C. 
i+*p . 
Hence it is evident that =~=nr is equal to i~\- 7 n—nx-\-m — n 
1 + .rl' 
