+

x (x−A) (x −2 A) (x−3 A) tan* A

A4

Now, these series being true for any value of A, are therefore true when A=0; in this case, cos A

(105.) Let e be the base of the Naperian system of logarithms, then it is well known that

if x be successively changed to x ✓-1 and

-x-1, we shall have,

(106.) By adding and subtracting the equations,

(1.), (2.), we have,

whatever be the number of arcs x, y......

By dividing equation (4.) by (3.), we have,

(107.) Dividing the second equation in (104.) by

numerator and denominator by cos x−√—1 sin x),

= (cos x + √ −1 sin x)2 + (cos x— √ −1 sin x)”

= cos n x+/−1 sin n x+cos n x-1 sin n x

(109.) To express the arc in terms of the tan

gent.

Since, e

cos x + √

sin x

x

=cos (1+1 tan x),

taking the Naperian logarithms,

x-1=log cos x+log (1+✔−1 tan x),

now, log (1+x) expressed in a series,