504 MR. J. W. L. GLAISHER ON THE THEORY OF ELLIPTIC FUNCTIONS, 
and the identity becomes 
tanh {x — ^) + tanh (tf-j-i^ + tanh (#_ f^j + tanh (x-\-%p)-\-&c. 
z being and v being We see from this investigation also that the left-hand side 
must consist of an even number of pairs of terms. 
As (35) is obtained by differentiating logarithmically the formula 
0 («+ K) =v / (§,y S ' 0 (»+K', K), 
it follows that (34) is a form of the identity that results from differentiating loga- 
rithmically 
e ~ r ' cos —— + 2e~^ cos-^-|-&c.}. 
The formula corresponding to (33) for the hyperbolic cotangent can be shown, by a 
process similar to that by which (33) was itself established, to be 
. • 2-kx . ■ 47 tX 
0 , 4 sin 4 sin 
coth a? + coth (x — ^) + coth (a?+/») + &c.=^+^-j cot ^+- ~& ^ ■+ - - &- + &c. [, (36) 
^ ^ ^ eT — 1 en—1 ’ 
2x 27rjsin 2z 
j«. ju. | sinh » 
sm4z i o 
sinh 2v~^~ C ' 
which holds good universally, on the same understanding, with regard to the number 
and order of the terms, as that which was found requisite for the truth of (33). 
§11. I now proceed to show how the identities which have been obtained in the pre- 
ceding sections by elliptic functions, or by Fourier’s theorem, can be deduced from the 
ordinary formulae for the cotangent and cosecant, viz. 
cotx~l+^+ x l % +~ 7r +^~+&c., (37) 
co^c •■'••• ( 38 ) 
by elementary algebra and trigonometry. 
Thus to prove (18) we have 
cosec (x J r ai) = — 7 —. — — - , — — — - V, -4- & c. , 
v ' > x-\ -ax x+ai—n x + ai + n 1 ’ 
cosec (x — ai )= — 7—. 4-&c. : 
v ' x + ai x — ai — n x—ai + w 1 ’ 
whence, by subtraction, 
2 1 — 2 — 7 — 3 — / — , "T 2“ 7 &c. = — cosec (x + ai) — cosec (x — ai)\. 
x l -\-a l {x-wf + a [x-\-Tty + a i 1 2 i l v 1 ' ' n 
