MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
159 
and we find from this the series 
X 
^+5 1 22“r'5 7,1 2 2^ 
2' 2'2‘ 
J 
Whence we find, putting f/j for 
n 
cos cos sin 0 cos ^+y— tan 
Next consider the symbolical equation 
(D— 1)(D — 3)(D — 5 )m— |«<V“ wi=0, where £“=<r; 
and assume as the transformed equation 
(D- 1)(D — 2)(D — 3)y— ^V“y=0. 
Then m=(D — 2)v, 
and ; 
where 1, a, |3 are the three cube roots of unity. 
Hence m=Ci( ^x^ — ^') + C2 — ■a?) + C3 ( —x)z^^. 
We must determine Cj, C^, C3 according to the series we have to sum. 
If 
w’e find 
8 pi 4(1+ — 3) p 4(1 'v/ 3) 
’~V’ V ’ V ’ 
A" j_ j 
rS J 1 -L ^ -I ^ f^L+_4- &C 
' Z 1 ® Z 1 9 ^ 
3*3*^ 
8 8 ■_Aff \/^ 
=^i{[/,x^—x)s'^'+j^(2i/jX^-\-x)s 2 COS-^f/^X 
3/*^ 
8 \/3 . 'v/3 
3|tx.' 
-T- xs 2 sm — [MX 
4 o * 
^TT TT 
Whence ££ €0S~^ ^ COS^ 0 
~2 *^“2 
cos|/«. cos ^ cos <p sin (^+9)+^+^— (tan 
' )=“ '^+ (6^;^+ 1 eos 
z 2 
(I-) 
H.) 
