of the Binomial Theorem. 
317 
Hence it is evident that if the series equal to i-\-x\" he 
multiplied by the series equal to i-\-x\ m , the product must be 
equal to the series which is equal to i-\-x\ nJrm . Now the two 
first mentioned series being multiplied into one another, and 
the parts being arranged according to the powers of x, the 
several products will stand as in the following representation. 
. , n — i . . n — t n — 2 
l-f+l = 1 nx -j- n . — — •£*+ 71 ■ 
2 
m — 1 
i-|-.rf=i -\-mx-\-m ,—^—x'-\-m. 
2 3 
m — 1 ?n — 2 
3 , 71 — i n — 2 n — 3 a , o 
x -f- n . . — — . — ^ X -\- &C 
x 3 -\ -m . 
234 
m — 1 in — 2 tn — 3 
o: 4 -j- &C. 
> » ft — I a I n— I 71 — 2 , . 
1 n . —^—x n . — — . — — X -f- » . 
11 — I 72 — 2 72 — 3 4 
mx - {- m . . n . 
n — 1 
x 3 -\-m . n . 
2 3 4 
?2—I « — 2 
w . 
. nx 3 -\-m . 
771 — I 772 — 2 , 
m . . x 4 -m . 
3 1 
2 
777 — I 
2 
■ . 7Z . 
3 
77 — I 
2 
777 — 2 
2 3 ‘ 2 3 
For the sake of reference hereafter let this be called 
multiplication A. 
Now with respect to the coefficients prefixed to the several 
powers of x, in the foregoing multiplication, two observations 
are to be made, by means of which the demonstration of the 
theorem may be extended to fractional exponents. 
In the first place, supposing n and m to be whole numbers, 
the sum of the coefficients prefixed to any individual power 
of x, in multiplication A, must be equal to the coefficient pre- 
fixed to the same power of x in the binomial series 1 -\-n-\-mx 
.r 4 -j- &c. 
.r 4 -|- &c. 
x 4 -h &c. 
. nx*-\- &c. 
-f- n-\-m 
n-H-ift—i 
x ZJ r n-\-m . 
72 — {- 777 — 1 77 + 777 — 2 3 
X —J— ll— I” 777 » 
2 ^ .r 4 ~f- &c. The certainty of this cir- 
cumstance rests partly on the 13th article, and partly on a 
77 + 777 — I 77 + 777 — 2 77 + 777 — 3 
