of the Binomial Theorem . 
325 
1 + - j| =^4-61”, by the general principles of involution ; 
and therefore a-f6l”=a”x : 1+7? ~ 4-7? . — ~ 4-7? . 2zzJL . 
' • a * 2 a r * 2 
7 +» • ^ • ==-* • ^ 7 4- &c. =*+»W’“ , +* • 
2 a 1 - 
n — 1 
2 3 4 
n— 1 k_2 ,3 n— 3 
« — 1 /„ « — 2 
2 
■ 6 ® « 
+7? . a n ~i+n .^i.a=l.a=l b'a n ~* + &c. 
2 3 1 2 3 4 
By article 19, 1 — nx-\-n . 2 — 1 x’ l —n . - — 1 . — - .r 3 -f* 
223 
. • ~~ • * r4 ““ & c - an d therefore as before, if —■ be equal 
to-r, a — b'(~a n —nba n 1 -\~n . 
h— i » n—2 
ha‘ 
n. 2 =i.?=l.?=Hfa H -+- & c. 
2 3 4 
i— « 
-;? 
By article 20, 1 -J--4 ”=1— 7?x-j-7? 7? 
M— 1 n—2, t n — 3 
6V -3 + 
!i±i w ~ 2 x s 
“J-7? . 
» 4- 1 « + * n -f 3 
2 ‘ ~T~ ' 4 
6 1 —» b 
x* — . See. and therefore if — be equal to x, 
1+ — =1 — 7Z 1-7? 
1 a \ a 1 
w-f 1 b x 
2 a % 
n + 3 b 4 
*!.+„. 2±I.!±i 
2 3 <z 3 1 2 3 
™ — &c. But by the general principles of involution 
— n 
iz”xH 
a I 
a 
—n 
-a 
x 1 4 — I =a-J-6l ; and therefore a-f-M 
■nba-n-'+n ,2±lA' a~“-*-n . i±i . b 1 a”" - 5 + « . 
' 2 23 ' 
2 3 4 
By article 21, 1 — x\~~ n ~i-\-nx-{-n . x 3 
+?? . 
n 4-1 «4-2 «4-3 
&c. ; and therefore if — be equal to x, 
1 — 
6 
«4- 1 b z 
2 a z 
~\-n . 
«4-i «4-2 6 3 
w4-3 6 4 
3 
P + n ‘ 
« 4- I « 4- 2 
-{- &c. But by the general principles of involution 
=a x 1 — > 1 61 and therefore a— 61 =ss 
fl XI-H 
a I 
