ME. J. W. L. GLAISHER ON THE THEORY OF ELLIPTIC FUNCTIONS. 
505 
Now 
whence 
e w_ e -m e ~m _ i 
= — 2 ie ni (1 +<? 2 “‘ -}-e 4 “+&c.) ; 
— cosec {x+ai)— cosec (x—ai)\=e xi ~ a -f-e 3 ^ -3 * -j-e 5xi ~ 5a +&c. 
4- e -*-« -(- e -3xi- 3 a e . -Ut- 5a _|_ &c 
and, on replacing x and a by — and we obtain the formula 
r + a 2 (x— jk,) 2 + « 2 (x + fx) 2 + a 2 ' (x — 2[x) 2 + a 2 ^ (x + 2/x) 2 + 
_L / 
■&c. 
Now from (38) 
7nr _2?[? 37ivr _ 
= — le ^cos — (-6 * cos — -f&c. 
' y y 
1 1 1 1 o 
— sec x— — * r~r + r~ + &c. 
x — pt x — fjr x + p r af — fir 
- La — . 3 ! 4_ _ &c . 
—x*-(br) 2 x*-(4*) 2 ^x 2 -(Pr)2 ’ 
whence, writing xi for x, 
sech x= 
and 
- sech (*-;»)= - (»_ J »)«+ ( 4x)*+(*-rt« + (fr)«~ (x-rt*+ (fr)* + &C - 
7T 3ir 
!+(i7r ) 2 -^ + (f w ) 2 -r^ + (^ )2 
37 r 
— sech (x -]-(«,)= ■ 
+ sech (x— 2 p,)= 
Lri2+ & C. 
(x + jx) 2 + (f7r) 2 ' (a? + ju,) 2 +(fir) 2 (a? + ju.) 2 +(fir) 
7T 37T 57T 
(x - 2/x) 2 + (f tt) 2 — (x — 2/x )' 2 + (pr) 2 (x — 2fx) 2 + (fir) ' 2 ‘ 
(39) 
&c. 
Adding these expressions together in columns, and transforming each column by 
(39), we find 
sech x— sech (x— p) — sech (x+^)+sech (x— 2^) + sech (x+2^) — &c. 
4-7T / 
' -A" 7TX , - 
3tt 2 
37TX 
e v cos — \-e 
2/x 
cos 
~ V- ' 
v y 
y 
47T l 
( - 7TX 
.?ZL 2 
3o rlc 
— 
6 ^ cos — + 6 
2/a 
cos — 
y ' 
y 
47 r/ 
^ 7TX 
15ir 2 
37 rx 
+ "7 ( 
6i ^ COS — + 6 
‘ 2/a 
COS 
y ' 
s. y 
y 
D7tX 
~ 
MDCCCLXXV. 
