of the Binomial Theorem . 309 
as it relates to the raising of integral powers, 1 proceed to de- 
monstrate, by the principles of multiplication, the most general 
case ; viz. that x 4 zV = x r -|- zx r 47 ‘ — 
-z x 
4, &c. This will clearly appear after it has been proved that 
JL 1 — 1 ~ — 1 - — 2 „ - — 1 
if the series x T -j — zx r 4 — x L — • % x r 4 — x 
• r 1 r 2 ' r 2 
x I— — z*x r &c. be multiplied by the series x r -j--r 
zx r + — x — 
% X r 
4 
•- 1 
X 1 z x r 
2 3 
+, &c. the product will be x r 4 -^^zx r 4 — 
*• + * 
n 4 - 1 
Z X 
+ 
W+I 
» + : 
«+* 
% X 
-3 
+ .&C. 
Or, which is the same thing, after it has been proved that if 
,i . ~ . n ~ * , n n-*-r % z , n n — r 
the series x r -\ ~ —zx r 4 7 x -77 % x + - x -77- x 
3 r J_ 
'3* x r 4 > &c. he multiplied by the series x r 4 7 
1 — r l — 2 r 1 — 3 r 
— : — 1 i i — r 2 — ~ 1 1 — r i — z r 3 — ~ — , 0 
zx r +-7X— Z X r +7X — X— ■ -z 3 x r -f, &c. 
« +1 
the product will be x r 4 ~XZ r + : ~X -- 2r 2T 
a + i— 2 r B + i— 3r 
x -7- + 2±i x X 2+i^r x~ ! ~+. See. 
' r 2 r 3 r 1 ’ 
14. Upon multiplying the two last series into one another, 
to obtain a foundation for the demonstration in view, the same 
powers of x and z, which arise in the multiplication, being 
mdccxcv. S s 
z r 3 
* + 1- 
m + i w+i— r 
