H4 Mr , Woodhouse on the Independence of the 
n — i 
a term &*-* , m = — . . 
«— -m c % z n—m c 
consequently. 
D w #*-* — - 
nb 
~ 
»*- ;,*-*+■ &c. 
a"— jS”=— __ 1— - 
2 ?z— m-fi ^ 
or In terms of ft, 
nb n . («-f i) («— 0 6 3 , n. («-j-i) («— Q (n-f 3) ft— 3 A* « 
2 T* I.2.3 - 9 2* « I.2.3.4.? *2^ €Xt ' # 
Hence, (sine nz ) =/> — n ' -— p 3 -{" &c« 
XX. Let ft be even, then term affected with x n in develope- 
mentof {lb7 -T^} = a *-^- 
Now ■ — - -~r 
I- I-J-/3.Z 
and the term affected 
1 — (« — j3) x — u@x z ' ! 
with x 71 ^ 1 in the developement of (i — bx — - cx 
is + d b n ~ z D a ti'— z c a + D 3 h n -3 r 3 + &c. 
term affected with in 6* x[ i — bx — cx + /3 = 6 s |- 
is 6* | b*— 1 -f D b n ^ z r-fD a b n ~-s c % ~\~ &c. | 
or in terms of n (r=i) 
is b'{tr~' +{n— s jfr-a+ (”-3)^-4 . gee. J 
Hence, since sine z 
and cosine £ = 
a-j-0 __ b 
b n 
2 y — i 2 y __] 
=P = =fc a— />»-■» 
a— iS A 1 ,, . a”-— S' 1 
-—x === - — == p sme ft£ — 
■* 2 * 
2 V— i 
2”~ s - (»— 3) «— 4 
1.2 
1 
X 
— p'l=tz p n ~ % =p= 2”~3. (ft Q)p n -3 dt 
or ==/*'! ^=' '{zp)*— 1 =f (ft — s) (2 ^)”“ 3 =*=&g.J> 
x p n ~~s =+= &c. the upper signs taking place, if ft is of the form 
ss (5 odd), the lower, if ft is of the form 45, s even or odd. 
If it is required to write the series in a reverse order, it is to 
be observed, that the series b n ~~ y + vh n ~ 9 + &c. terminates at a 
terml> w , when m == — — 1; consequently, igT 
and a— p=b'{-4 + 9’- &Z + &c. } 
