Mr, Herschel on various points of Analysis, 451 
The usual exponential expression for tan. reduced into two 
fractions, gives 
tan. 0 = 
v/— I + v/ZTi ; 
Thus, the first member of (12) becomes 
tan. 9 
g J (— 
2 n—i 
or, k <1 
2n— I 
tan. 20 , tan. 39 
- 
» 3 2n_i 
2 / 1 - 
(13) 
&c. 
In order to obtain the coefFicients a , a a , in the 
i’ y zn — i’ 
second, we have, 
/(^ ) = 
v/~ I -i 
z=z a a + &c. 
0*1 *2 • 
Now/Ce^) may also be thrown into the form 
V — \ -j. s/— I 4- 8—^ 
which is the same as 
^ tnn. ( 
+ i 
VET 
+ * 
t 
itan. + X V'--i_±sec. 
Thus the even values of are given by the developement 
of sec. Slid the odd by that of tan. [^/=] hence 
it is easy to see that 
,2r-I ^ 
2X 1 
• 
%/__j 1.2 3 .... (2 at) 
. B 
2;r— I 
(> 4 ) 
and 
1 
^zx 
12... ( 2 jr) . z^^ + ‘ ] 
&C. 
}— &c.|: (15) 
but a 
general value of 
immediate consideralion 
zx 
2X 
We know that 
