MR. J. W. L. G-LAISHER ON THE THEORY OF ELLIPTIC FUNCTIONS. 
503 
by 2 also ; while if we replace x by x — (a both sides are diminished by 2 ; so that (33) is 
true universally for all values of x, on the understanding that the left-hand side is 
tanh #+{ tanh (#— ^)-ftanh (x-\-[/>)\ + {tanh (x— 2^) + tanh {x-\-2^)\ -f-&c., 
viz. that after the first term the series is to proceed by pairs of terms ; so that for every 
term tanh (x+nt/j) there is also a term tanh(^+w^), and the whole number of terms 
included is uneven. Thus for x=^(jij the series is 
tanh -J- { — tanh + tanh f ^ \ + { — tanh f ^ + tanh \ + &c. , 
the value of which is unity ; and not 
{ tanh — tanh ( + { tanh § p, — tanh f -{- &c. , 
which is equal to zero. 
If we write for x, and suppose the terms arranged in pairs from the begin- 
ning, we find 
{ tanh (^-(-^j-f-tanh (x — ( fi- ) tanh tanh (^-f^)|--j-&c. 
2x 
^ jcosech — 
. 2%x . 2tt 2 . 4nx 0 ) 
sm — cosech — sm — + &c. y 
p P , a p ) 
(34) 
as the unity which is introduced on the right-hand side by the change is cancelled by 
the unity on the left-hand side, which results from the supposition that the number of 
terms is even. 
The last equation is, in fact, the relation 
iZ («+K)=^ 7 +Z(u+K', V) 
(35) 
(Fundamenta Nova, p. 165, and Dueege, § 69) ; for 
2 ^ C q (ft | 
Z (u)= k|]Y ^2 sin 2x-\-j Z ^ i sin sin 6^ + &c.| ; 
so that (35) becomes 
2<7ri ( n (ft (ft 1 
jH - sin 2 xi+Y^ sin sin 6 ^‘+ &c - [ 
=2&'+£{“T=^ sin sin 4z ~ &c - }, 
of which the left-hand side 
=jl{(e 2x -e~ 2x )(q+f+q 5 -\r&c.)-(e 4x -e- 4x )(q 2 +q 6 +q 10 +&c.)+&c.} 
7r ( qe- x qe~ 2x q 3 e 2x 
Kjl + g'e 2 -* 1 ^-qe^ 21 ' 1 +§ 3 e 2jr 
1 — qe 2x 1 —qe~ 2x 1 —q 3 e 
1 + qe 2x 1 + qe~ 2x ^ 1 + q 3 e 
( fe~ 
r^4-&C. 
1 + (fe 
*-r+& &c U 
