of the Binomial Theorem . 
313 
xi — r x 1 — 2 r, 1 x 1 — rx 1 — 2 r x 1 — 3r, &c. More- 
over, put «= 1, /3 = ix2, 7= 1x2x3, 1 x 2 x 3 x 4, &c. 
and then the multiplicand, multiplier, and products will stand 
in the following manner, the powers of x and z being omitted. 
^r + ^r+ 
' + 17“ + 77“ + + 7 77 + ’ &c - 
, A B C , 
^ ■ *r "•* .Sr* ■ ^r 3 • 
, a 1 b. . 
1 + Tr + — + 
V+-£-+. &C - 
, A B . 
1 + Tr + TF + 
J 3 r 2 
a . a A 
» r * aar 
b 
Br* 
1 D 1 I 1 Jf t_ 
t- ^r 4 "T- £ r 5 "T £r 6 “T „r 7 ■" : 
_C 
yj 
. a B . «C , « D , a E 
1 A” a|3r 3 ay.r 4 * aJr 5 * a. c T 
JL UL _L iJL _L _^£. 1 IR 
• 0 a r 3 "r 00 r 4 > 0yr 5 + 
&C. 
+ 
r r 3 
•£ + 
0 £J - 3 
<rD 
&C. 
/ 3 ( 3 r 4 
cA 4. L _i£_ _i_ _L±L 4. gr C 
1 y/3r 5 yyr 6 * y^r 7 
y ar 4 
<* 1 _dA 1 rfB ' 0c . 
£r 4 “t" J ar s 1 407 s 1 ^77 “T» ^ C “ 
e A 
e B 
&C„ 
T 7 r + 7 ^ + ^+ ! 
TF + {7? +’ &c ° 
~r +, &c. 
Now the object in view, with respect to the coefficients, is 
to prove that the perpendicular lines of products will be, be- 
ginning at 1 and reckoning towards the right hand, equal to 
n - f 1 
m+i n -f 1 — r tt-fi 
r * 2 r * r 
»+ 1 — r m-M— zr 
~ x “77-’ 
&c. 
respec- 
tively : and this will be fully demonstrated when we have 
proved that all the terms of products in any perpendicular line, 
in which the exponent of r in the denominators is if, being 
