analytical and geometrical Methods of Investigation, i o g 
representation. I think, therefore, the analytical demonstration 
in which the symbol \/ — 1 is introduced, (for the cosine of an 
arc cannot be adequately and abridgedly represented in terms 
of the arc, except by means of the symbol \/ — i,) to be the 
most simple and direct that can be exhibited. I have endea- 
voured, in a former paper, to shew that demonstration with 
such symbols as V — ~i may be strict and rigorous. 
XVII. One or two more instances of the advantage accruing 
to calculation, from giving to quantities in analytical investiga- 
tion their true analytical representation, I now offer, in the de- 
monstrations of the series for the chord of the supplement of a 
multiple arc, in terms of the chord of the supplement of the 
simple arc, for the sine of the multiple arc, &c. 
Chord 27 r — % = ( V — 1 ) 1 
( 27 T z) 
V—i 
■(2 K—Z) 
7 V— i 
V — x 
— z 
v~t 
-f- e 2 ", since s 
Again, chord ( 27 r — nz) — e 
£ 
l 
= I/! 
■1, ands vi/ x — _V__- 
nz — nz 
~ v - ! + 
v:=7 
. Let s 2 
s 2 =/3 «/ 3 = 1 ; what we have to do then, is to find 
a"+j 3 " in terms of and, for facility of computation, a 
new mode of notation may be advantageously introduced, which 
requires a brief explanation only.* 
* I had obtained the forms for chords nz, &c„ given in the following pages, by 
actually expressing in terms of n and b, the coefficient of x n , in the developement of 
the trinomial ^ i bx-\-x z j' n , when the very admirable work of Arbogast, Du 
Calcul des Derivations, came to my hands. The great simplicity and convenience of 
his notation have caused me to adopt it, although it does not harm onize well with the 
fluxionary notation which I have employed in the present Paper, 
