512 
ME. J. W. L. GLAISHEE ON THE THEOET OE ELLIPTIC FUNCTIONS. 
whence, since the first series on the right hand side=^T, when x is positive and less 
than /A, 
sing sin3z jx ( cosh (ju.— 2x) cosh 2 ([x.— 2x) 
sinh^ v ' sinh-|v~*” C 2ir | cosh^ ' cosh2j& 
— &c. 
) 
i 
and 
tanh x - tanh (x-p)- tanh (x + p) + &c. = 1 - 2 C0 ^ h ^ x) + 2 ~ ~ &c. 
The other transformations to which the method of this section leads are 
coth x — coth (x—p)— coth(# + p) + &c. = 1 + 2 C0S |? 0 ^ h ^ + 2 -" cosh* 2j/~ + &c ~’ 
cosech # — cosech (x—p)— cosech (a? ■ +/x) + &c. = 2 + 2 ^ coih|^ ^ + &c -’ 
cosech x + cosech (x — p ) + cosech (#+,«,)-}- &c. = 
secha;+ sech(#— sech(a:+/x)+&c.= 
s inhft^j sinh3(^-x) 
sinhfju, 1 smh-iju. 1 ’ 
cosh (If*— a?) O cosh 3 (&*—#) 
sinh^ju, sinhfju, ' C ‘ 
which can be readily verified by ordinary algebra in the manner explained above. In 
all these identities x must be positive and less than 
§ 16. It only remains to apply the methods of §§ 11 and 15 to the identities (33) and 
(36), which differ from the others by relating to non-periodic functions. Employing 
the method of §11, we have 
tanh x— 
** +($*)* 
2# 
+ * 2 +(f 
+ ^-+(142 + &C., 
, ,, , 2 (d7— /*) 2 ( x —[ l ) 2 
tanh(# + ( a? _ ft )2 + (| J!r )2+( a? _ p )2 + (| w )2+&c. : 
j_ , / , X 2(a: + /i) , 2(a? + ft) , 2(x + p) , 0 
*«nl‘(a+rt_ (#+/l)1+{W ,-l- (le+(i) , + ( )4 + (lt+rtS+ , (WS +&c. 
whence 
tanh,r+tanh(;r— ^) + tanh(#+^)H-&c.=^ j# -1 ' wn2z-\-e~ 2v sin4s+&c.| 
-J- 4 “ le -3 " sin 254-^ _6 ‘' sin 4z-f-&c.j 
sin2z4-e _10,, sin4s-f-&c. j- 
+ 
= j (rfps sin 2z + sin 4z + &c -) 
27r/sin2g , sin4g ( Q ^ 
— 7 ^ihdTv + + &C - 
( 48 ) 
