e 
Fae Demonstration of the Binomial Theorem. 
ad infin. in which p=m-+n which is the known formula for 
the expansion of a Binomial. Again, by dividing both sides 
m ml (n—1) as a 
of the equation wigeeg hak Ce ee ax? > 
&c. ad infin. - = after the manner of common — 
I have (atx) oa es saab eee 2? + 
Xe. ad infin. in which m, is changed into rey, divide by 
a+< again, and m—1 will be changed into m—2, and so on 
to m—n, if n tg the number of divisions by a+. hence 
Ihave (ata) = =a+ m—n a SF (in n) x (m—n—1) 423-4 
Ee te, 1x2 
&c. ad infin. or (ets)=at (7) 4 a ‘eas (<x tapers ; 
ca * Hop x(=p— ae a + &c. ad infin. If p 
~ 1x2 
=n—m, which, is the expansion of a Binomial in the case 
of the exponent negative, but integral, and it is of the sam 
form, as in the case of the exponent positive, only p jalimad 
of being positive, is is here every where negative. Twill now 
+pq—1 
nents. To this end, I take ries et iepen ite 
—- 
shpq—2 P +p 
CEept Cee q—1)¢4 «?+ &c. ad infin. and (a+zr)= 
BE Oe 
ae = (tee “+ @4)x ivr -p—l)aa 4 end. jock: 
tp and g teed both i wend These formule are = 
ly true from what I have previously shown. Now << 
PY 
is the gth rootof (4+2)— asis well known. Hence to derive 
