sums of several classes of infinite series. 
and when n is an odd number, 
2 57 
= A 
—i 
X X 
(sin 0 )" 
—2 
X 
"" + A (sin'2~* + A 
x 5 — x 
(sin 30)'* 
+ &C. 
From these expressions it appears, that the reason why we 
have succeeded in the integrations is, because we had so as- 
sumed %{/, that the sum, ^ v zz+n +i!/v m ” ' is a constant quantity ; 
the same success must follow whenever this condition is ful- 
filled : and hence, we have a method of discovering the sums 
of a great variety of series, containing the powers of the sines 
of arcs in arithmetical progression in their denominators, by 
solving the functional equation -{- ipv~ m2z ~ n =c. This 
is fortunately one of a class whose general solution I have 
arrived at ;* it is 
2 z + n 
— zz—n 
2 z+n cpv 
~.. zz + n i —2 Z- 
<pv +<pv 
— or ^x = 
c<px 
q> x + <p — 
X 
In the example I have employed n was supposed equal to 
unity ; if this is not the case, we should have found 
( 2 V— 1 Y f 2± ~* _i_ j3±x ' 3 - & c 
V * V 1 ) ( S i n e)« (sin 20)” ^ (sin 3 0)» <XL * 
If in the functional equation we put c = i, and <px = tan' x , 
2 ml 
then we have -\x~- tan x, and 
( 2 V-0”5”(0 = |{T7i^ 
x 3 ±x* 
+ 
X s ±x 
p &C.} 
(sin 0)” 3 (sin 3 0) n • 5 (sin 50)” 
the upper or under sign being used as n, is even or odd; if 
n= 1, the constant is zero; and we have 
It log X X— X 1 
20 I sill 0 
x 3 —x X 3 — X Q_ 
3 sin 3 0 "T" 5 s in 
(9) 
MDCCCXIX. 
* See Philosophical Transactions for 1817, p. 202. 
LI 
