of the Binomial Theorem. 315 
The first of the two conti- 
guous lines multiplied as men- 
tioned above. 
F 
G 
X p 
+ 
a = 
= G + flf F 
a E 
F 
X E 
+ 
b _ 
a ~ 
-a F -f 6 E 
b D 
E 
X D 
+ 
c 
T = 
= b E -f r D 
c C 
D 
x r 
+ 
d _ 
c 
= cB + dC 
d B 
C 
x T 
+ 
e __ 
7 = 
= d C + *B 
£ A 
B 
X A 
+ 
/ _ 
= rB +/A 
/ 
A 
x T 
+ 
g _ 
/ ■ 
=/A + £ 
The last of the two contigu- 
ous lines as mentioned above. 
G 
a F 
b E 
cB 
d C 
e B 
/A 
g 
The proper denominators being annexed to these terms, 
and v being put for t 1, it now remains to be proved that 
G-J-aF , dF-f-^E , 6E + cD , cD-j-dC , -d C-\-eB . eB-{-fA ( 
$Br 6 xvr > 
£r 6 xv r 
; r 6 XV r T /3 $ r 6 xv r * y yr 6 xvr 
ia.r 6 xvr 
f A +g 1 a F , &E , c D , d C . e B , /A ■ g 
^r 6 xvr » r 1 ' »£r 7 ■" $ £ r i T y $ r i r i T s @ r i T ^a.r\ ' » r 7 * 
18. The relative values, therefore, of a,, (3, y, &c. next claim 
our attention ; and from the nature of the series, — — r — 
. 7 V V — I 
v — 3 
y __ q p __ x 
1. Also 1 = 
cc,-j— — y,j= 2 ,-^ = e, j = C As the powers 
of r in the equation, asserted in the end of the last article, are the 
same in all the terms, they may be neglected ; the only thing 
necessary is to reduce as, ( 33 , yy, $( 3 , s a, £, the denomina- 
tors of the first side, to ^ , a £ /3 s, y $y, e [3, £ a, 17 , the de- 
