,0 
the tynomial theorem . 
M. 
s'A a 2m — 1 4-2" A a zw ” 2 ' 
+ 
'A 7 2m-; 
1 5 
p $’"Aa zm — 3 ' 
4-4 LAa zm —4*\ 
-2 "Aaz m ~3 . 
>x 
4-3" / Aa 2Wl — 4, 
Jf 
niii; 
3 rtryr 
333 
" fr> “ 
+ 
+ 
bv putting for *■ its value l + And by comparing the co- 
efficients of the different po wers of x there arise 'A='A ; 'A A 
= »"A + 'A; 'A "A ay | '"A + 2 "A ; 'A '"A = 4 ""A + 3 '"A ; 
and so on; whence "A ses W A = — — — " W A= 
!A£A_J] ; and so on. 
bomnggB sd 
ir. sl 
I ft i Ir 4 
1 odl jsnod 
• ii? ea*® _ A 'yo, 
Such is the law by d/hich the coefficients are derived from 
each other, whatever be the value of m ; it remains to find 'A ; 
but I shall first observe, that if, instead of the assumed form 
of the expansion, we had made 
(a + x) m =za m 4- r A a™—'x + "Ax* + 1,1 Ax 5 + as some do, our 
demonstration would have succeeded exactly the same ; be- 
cause the exponent (m) and the first term (a) of the binomials 
are the same in all the three powers employed. 
We have already seen that 'A = m, if m be a whole posi- 
tive number ; or that (a + x) m = a m -f* ; and from 
the value of 'A in this one case its value in all the others is 
easily discovered : thus, let 
J :x £4 
* 1 A-x 
• nr 
A' 
(a -j- x)™~ a m -{-'Aa™ x +, the m tb power of this is a + x; 
( _L J j n Y LL _ m ~ m X - 1 i 
\a m 4-'Aa™~ 
but 
- -Lodi 1 ™ — i8udi^njvfiH( , 
x-\- ) =a m 4-m Aa m x# w x-\-=a-\-m Aat4- 
noiiBlipe ? tn m mad! oliniJadua ol jua 1 
but a + m s Ax + = a+ x, consequently 'A = ~; and 
jl A A.z luZ . . 
( a 4- x')™ =a m 4~ m a m x 4 -, next raise this to the n tb power, 
1 • • . • • . 
m 
. 
>0 v-.: 
n being a positive integer. 
A- 
-nts 
-in 
\n 
n 
n — x 
-m 
