analytical and geometrical Methods of Investigation. 115 
or in terms of n 
1) y f n . («* — i) (n % ~q) Ls 
i 6 s == &c. 
2M . 2 0 3 2+ . I . 2 . 3 . 4 . 5 
Consequently, 
cos. «* = £(=•=«& ^ «J£=ll» =*= ”•(«•—) (^-9) ii 1 
2 2 l ^2.3 Z z 1.2. 3.4.5 2+ / 
= =*= 
A l • 2 • 3 r 1.2. 3. 4. 5* 
Where the upper or lower sign is to be used, as n is of the form 
(45 + ij, 01*45 + 3. 
XIX. Again, sine z ~ (s/^i)" 1 e — 
sine nz = {2/ — 1 j‘”“ f ( — e — 
it is necessary to find a? — jQ" in terms of & — 
Let & be odd, 
then term affected with a;” in developeinent of | } 
— nnd 1 1 1 — 2 “" 2 — fr* 
^ J I— ax * i + ^ I — (« — ( 3 ) x—a@x z X — — c.r l ' ? 
and the term affected with x n in the developement of (2 — bx) 
(1 ~ 6* - cxT 1 ==^+” + ”? &c. 
or in terms of n (c = 1) 
= &’ + ?* 6 s ' 2 + _ &c . 
but sine 2 = ~^== = /> .*. (6)"— {2\Z’^ip) r = z^ = . % n s/™\p\. 
where the upper or lower sign is to be used, according as is 
of the form 45 + 1 , or 4 s + 3. Hence, 
a "— 0 " 
— .V 
S— i 
; 8—2 
^*= 5 = 2 * S .w / 3 ‘ =±=2 
R ““ S 0 n -( n — 3 ) («— 4 ) j .*~4 
i . 2 . 3 ‘ i 3 
sine 722= 
&c. 
If it is required to write the series in a reverse order, it is to 
be observed, that the series b n + d & c. terminates at 
MDCCCIL 
