sums of several classes of infinite series. 
265 
and co-tangents in arithmetical progression, thus (1,9) com- 
bined with (a) will produce 
(sin 0 ) a * (sin 2 0 ) a * , (sin 30) a4 
-r “ 
(COS 0 ) ra (cos 20 )" 1 (cos 30 ) n 
But the left side of this equation is equal to ~ (l — i)*= o. 
Hence 
See. 
_ (sin 0) 5 
(sin 20) ai 
(cos 20)" "T" 
(sin 30) 1{ 
&C. 
(2,8) 
(cos 0) n (cos 20)" ~ (COS 30) n 
And if n = 2k, this produces a series of tangents 
0 = (tan Qy * — - (tan 2 0)®*+ (tan &c. (2,9) 
By means of the theorems already referred to, we may in- 
troduce into the numerators of each term of the series (2,4) 
the even powers of the sines of the same arcs whose co-sines 
occur in the denominator : putting l = ~ , we shall have 
£ { C 2k,n — 7 C 2k,n- 2+ lj TT C 2k,n~4~- &C * } = 
(sin 0) a? 
i w (cos 0) n 
(sin 20) 24 
+ 
(sin 30 )** 
3 a4 (cos 30 )" 
~ &c. (3,1) 
2 24 (COS 20) n 
And if 11 = 2/, this becomes 
2" \. C 2k,zl 7 C 2k,2l— 2~^~ 1.2 C 2k,2l—\ ^ C * } 
(tan 0) az (tan 20 ) %l , (tan 30) a; 
• ^ H jol 
If we call the sum of the series (2,5) A^ w , and if we apply 
to it the formula (6), we shall have 
(3.2) 
1.1—1 
_ a | li> ' 
i k,n — 2 * 1. 
A, — &C. = 
4 
(sin 0) 2 ^+ I 
l z *+V;os0) K 
(sin 20) 2 ^ + 4 (sin 30) 2t “f' 
+ 
2 2/c+ i( C0S 2 g)« 3 Z ^+ 1 (cos 30) ?i 
And if w = 2/ + 1 , this becomes 
A, . . — — A, . + &c. = 
&c. (3.3 
(tan Ci) 3/ + 1 (tan i 9 ) 8< 4 ' , (tan 36)^ + : 
,2fe+I 
M m 
2 2& -j~ I 
&c. 
(3.4) 
MDCCCXIX. 
