108 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
Expanding cosh a{x — ir) as a Fourier Series in the region 0 <^r< 27 r, 
we easily obtain 
cosh a{x — 7t) = sinh air 
The limiting case where gives 
00 
2 
-1 CO -1 
1 ^ cos nx 
cos nx _{x- 7t)2 
4 "6 
• (65) 
• (66) 
+ GO 
••• 2 
/cos {2n+ 1)(^ 
cos ^ 
+ CO 
2 
cos (2n - 1)(^' 
11 ^ cos 0' 
_t^/cos27^<^ ^^/cos2w(^' ^i^/sin 2w</) tan ^ — sin 2 ? 2 ^' tan 
m2 m2 
_ 9 
(2<^ — x)^ x^ 
- 2 
(2<^^ — x)^ x^ 
L 4 6_ 
[4 ej 
= (c/) + <^' - 7 t)(<^ -</)') . . (67) 
Differentiating (65) with respect to a, we find 
00 
cos nx 
1 7T cosh a((T - 7 t) ttx sinha(x-Tr) cosha^r 
+ — +— o— (68) 
‘^(a^ + w2)2 2a“^ 4a^ sinhaTr 4a‘^ ■ sinh ^tt sinh^avr 
00 9 9 00 00 
XT' cos « 9 cos 
m2 
/ m2 
7^ + a-)^ 
7T cosh a((T - 7 t) 1 1 7T cosh a{x - tt) ttx sinh a{x - tt) cosh ax 
2a sinh ax 2a^ 2a sinh ai? 2 sinh ax 2 sinh^ ax 
1 x^ cosh air xx sinh a{x - tt) 
'2a? 2 sinh^ ax 2 sinh ax 
(69) 
on using equations (65) and (68). 
Putting x = 0, we obtain finally 
2 a^ 
(d^ + /^^)‘^ sinh^ ax 
(70) 
But substituting — for a, and 2x for x in equation (65), we obtain 
cos 2ir cos 4x 
a^ + 2^ ■ a^ + 42 
and subtracting from (65) we find 
cosh a X - - 
X 
4 a 
. ■, ax 
smh — 
l)ir 
X cosh a(ir - x) 
cosh a( ir - - 1 
1 sinh a( X - - ) 
X V 2^ 
_ X V 2/ 
■1)2- 
2a sinh ax 
4a . 1 ax 
smh — 
4a 1 ax 
cosh — 
2 
2 
(71) 
( 72 ) 
