506 MR. J. W. L. GLAISHER ON THE THEORY OF ELLIPTIC FUNCTIONS. 
which, after summation of the columns, 
e v 71-2? , e 2,1 
— -4 cos — -) 
l+e~J ^ l+e~ 
Bwa? . e 5i tx 
COS rrj COS 
1+e n 
=?( 
sech 
TtX , , 37T 2 37T2? , 5tt 2 5wa? 
cos — f- sech cos h sech - 5 — cos 
? +&c -) 
+ &C. 
ift ft ■ Zffc ft ■ Zft ft 
which is the identity (18), that was in § 6 deduced from the formula 
cos am w=sec am (ui, k'), 
and in § 8 from the integral 
j sech x cos nx dx=^ sech 
§ 12. The other identities, (19) to (23), admit of being demonstrated in exactly the 
same way. The formulae of transformation, similar to (39), that are required are 
a?— ft 
X + fX. 
a? 2 + a 2 (a; — ft) 2 + a 2 (a?+/t) 2 + a 
(T-$*+ « 2+ ~+a*+ &C - =j! ( X + 2 *~ T 
, _I® . ?ra? . 37ra? 
+&c.=— (e sin — ^ sm — +&c. 
ft ft 
27ra? „ 4vra? 
cos K 2 e * cos h&c. 
ft 1 ft 1 
! + / 
a? — ft 
a? -{-ft 
,+&c.=^(^ 
. 2?ra? . 4?ra? \ 
sm ^ sm |-&c. ), 
<*?‘4-«‘ ' (a? -ft) 2 + a 2 ' (a? -f ft) 2 + < 
the first resulting from cosec (x-\-ai)+ cosec ( x — ai ), and the other two from cot ( x-\-ai ) 
+cot ( x—ai ). The following expressions, which are analogous to that used for sech x 
in the last section, are also needed : — - 
tanh x= -s 
2 x 
! + 
! + a7 2 + ^)2 + &C -, 
2a? 
1 2a? 
COth X — “J~ o o+* 2 Trt \2 “1“ 2 To \2+* &C. 5 
+ 7T 2 ^# 2 + (2tt) 2 ' # 2 + (3?t) 2 ' 
1 2# 
cosech <r= 5 - 7 — 2 + 
nf* nr>* -L- nr* 1 
+ 7r 2_r a? 2 +(27r) 2 a' 2 + (3 tt) 5 
; + &C., 
all of which follow from (37) and (38) at once in the same way as that by which the 
formula for sech a 1 was obtained. 
Only one point calls for notice in these demonstrations, viz. in the proof of (20) we 
find 
sech x -f- sech ( x — ju,) + sech (a+jM/)-J-&c. 
27 r 
t* 
l-\-2e ./* cos^--j- 2 e 
271-a? „ 
cos + 2 i 
2tt2? 0 4 %x 
COS + 2(3 “ COS b&c. 
P ¥■ 
