analytical and geometrical Methods of Investigation. i ig 
= 6“ 
= b '{ 
nb 
2 
nb 
+ 
+ 
( n . n In 
1.2.3 
n (n z — 4) b * 
1.2.4 ' 2 3 
b 3 -j~ See 
■} 
— See . } 
consequently, sine 7*3 or a ” ' 
2V— 1 
n . (n*— 4) 
^ + 
« . («*— 4) n e — 9) s 
/> S — &C. j 
x.2.3 * s 1.2.34.5 
XXL The sine rc# (0 even) may be expressed by series, in 
terms of the cosine of % ; 
thus, -i ' 
9 f 
I'—UX 
I — 
+13 x+cx z 
and, equating the terms affected with x n in each developement s 
we shall have 
sin. nz=p{(ap' (a/J—s 4 fc rb ( 2 /)»-i_&c. i 
when n is even, a series may be found for sin. nz in terms of 
p (sin. z) only; but this series will not terminate as all the 
foregoing series do. 
To find this series, expand */( 1 — -/>*) =/>* into a series, 
if — D if/) 8 + if p* — &C. 
then sin. nz— j 1 — di I/’+b' j x | ^ »•(»*-»•) .., , See. 
— np+ Ap’+ A, p’+ A,,/ 4 &c. 
in which series, the law of the coefficients, or a general expression 
for a may be found. But it cannot now be done, without too 
long a digression from the present objects of inquiry. 
From what has been done, the series* of the chord of the 
• Demonstrations of these forms have been given by reversion of series, and by 
induction ; which demonstrations are imperfect, since they do not exhibit the general 
law of the coefficients. See De Moivre Miscell. analytica. Epistola de Cotesii 
Inventis, &c. Newtons Opera omnia, p, 306. Euler in Analyt. inf. Cap. 14. 
Waring has deduced the chord of the supplement of a multiple arc, in terms of the 
chord of the supplement of the simple arc, from his theorem for the powers of roots 1 
