112 
Mr. Woodhouse on the Independence of the 
m -f- i ) ^ 2 * 
pm m (n — m ) n — m — i) (n — m — z ) [n - 
C I • 2 » ^ •••••«»•••••• ?)2 
a” 4- = 6” — zz b n - z + -- -~ 3 6* -4 — 6»-6 i & Co 
the series for the chord of the supplement of a multiple arc, in 
terms of the chord (&) of the supplement of the simple arc. 
XVIII. Similar series may be found for the sines and cosines 
of multiple arcs; thus, 
f 
cos.% 
Now, a 
v. cos. n% 
2 — ^ z zV — 1 -|- g -~ ri_i cos< 
== g xA/ ~ cc n — t n al/: 
a. -f j 3 = Z p = h, 
cc n +i 3 n 
nz = 2 ” 1 1 -j- e" 
Let cos. z = p 3 
,.zV: 
i . [ 2 n p n — n . 2” 2 p n ~°- -j- ” ' n 2 n — 4 p n ~±. 
&c. 
__ 2^-1 __ w . 3 k -3 pn—2 _{_ hi 3 2 «_ 5 pn-± __ & c 
or = 4 { {spy — ?z . (%py ~~ 2 + ( 2 py~~ i — &c. } 
Suppose it were required to write the series in an inverse 
order: let zz be even, then the series b n 
n 
n — i 
d b n 1 &c. terrni- 
. , , n J n — m n j n 
nates at a term y b , m = , and ~— 
77 — -TO C J 2 77 — 777 
2, and 
a n + &”= =t= 2 
or, in terms of n, 
n — 777 -}- I c 
jy m ~ l pi—m+1 
n 
n — m-\-z c 
D”- 2 6"-”+ 2 — &c. 
. n 
1 . 2.2 
n n n 
71 ' 2 * 2 ‘ 1 ’ 2 
i . 2 . 3 . 4 
6 4 == &C. 
t n + / 3 n 
77 * rC . (/ 7 a — 4 ) p 4 
I .2 
I . 2 . 3.4 
&c. 
Consequently, cos. zz^ = 
Where the upper or lower sign takes place, as n is of the form 
(s an even or odd number), or 2 s, ( s an odd number) ; 
, ... n m,® i n — m 
let n be odd, then the series terminates at a term y 0 
m 
n- 
: 1 n W® 1 n — m 7 
-, and .*. - — - . V b = nb. 
and a? 4- 
n — m c 
n 
nb 
n — m-\-\ c 
jf - 1 
&c. 
