498 ME. J. W. L. (tLAISHEE ON THE THEOEY OF ELLIPTIC FUNCTIONS. 
cally our knowledge that sin 0+sin 20-J-&C. and sin 0+sin 30 + &C. are the Fourier’s- 
theorem equivalents of cot and ^ cosec d would be sufficient to leave no doubt of the 
accuracy of the formulae so obtained. 
In regard to the other two integrals required for (19) and (22), viz. 
| tanh x sin nx dx and j coth x sin nx dx, 
Jo Jo 
it is to be observed that, stated in this form, their values are indeterminate ; for the 
former 
and the latter 
=J 0 ( J - <®tt) sinnxdx ’ 
=1 ( 1 +^) 
both of which involve cos go . But in point of fact for our purpose the co of the limit 
of the integral is not arbitrary, but is to be of the form (m-\- 1)t, the lower limit being 
—mir (or if we replace sin nx by sin the limits are (m+l)^ and — m[h). Taking 
then m infinite, 
tanh x sin nxdx 
Jo 
!~ cos nx~ 
l(B+I)>r 
—2< 
f n 
L _ ~. 
L 
[n* + 2 2 
r cos nx~j 
(m+l)7r 
n( 
1 
L~~J 
0 
(*»)*+ 
smnx dx — 2 
„2 ,_a2~\~~„ 
OIU /tot/ 7 
e**+l dX 
— &C.| 
Similarly 
and therefore 
. r cos nx~ 1 S m+1 ' )v 1 , , n% 
= T~\ --+2 c osech T - 
L_ n Jo n L 
tanh x sin nx dx—\ tanh x sin nx dx 
J — W7T J 0 
r cos>nx~\ m * 1 . v , n% 
= k-cosech-^5 
L » J 0 n 2 2 
m . , , I - cosraaT! (m+1),r , r cos nx~\ 
tanh x sm nxdx=\ - + — 
L n I n J« 
2 nir 
•-+9rcosech ^ 
=t cosech 
(30) 
whether m be even or uneven, if n is uneven ; whence the result in (19) follows directly. 
