MB,. J. W. L. GLAISHEE ON THE THEOEY OF ELLIPTIC FUNCTIONS. 509 
cos am M=Jgj eos #+y ^3 cos 3#+y^5COs5;r+&c.j 
■*K' 
g-(l-g) 
gKi-g 3 ) 
3 + &C. 
1 — 2 q cos 2x + q 2 1 — 2 q 3 cos 2x + q 6 
U7(R-^J-G«) +&c -i 
— AKO £ __ x _.__ w _ T1 , 
''7'^ + r * + r Vjt J r* -\-r 
= 2 ^/| sec h 2 — scch (z— v)— sech ( 2 +j')-f&c.[ 
= 2AK' / sec ^ 2 — 4 cos ^ 
.( 5 
ycosh 2 
cosh i 
cosh 2v 
+ cosh 2» cosh 2 z + cosh 4v 
+&c.)} 
= 2M'{ sech 2 “IT r cosh ^+IT^ cosh 32 — &c.| ; 
while x, z , //-, y being any four quantities subject to the relations 
(JjV=tt 2 , z=~( whence 
the identities are : — 
sech # — sech (x—p)— sech (x + p ) + sech (x— 2^) + sech (x + 2/ca) — & c. 
=sech,r— 4 cosh # 
cosh - 
cosh 2 ju. 
( cosh 2# + cosh 2ft cosh 2x + cosh 4/x 
4 cosh x , 4 cosh 3x 
+ &C.1 
:S e ch*-^ Tr + e „ +1 
cosh 
— &C. 
2%( cosh z cosh 3z cosh 5z _ | 
p | cosh ^v + cosh fv cosh -fv C ’ j 
sinh 
+&C+. 
sinh fv 
[sin 2 ,? + sinh 2 £v sin 2 ^ + sinh 2 -|v 
Another form will also be given in the next section. It is scarcely necessary to 
observe that corresponding formulse and identities exist for sin am u, A am u, cosec am u, 
sin am u „ 
a , &c. 
A am u 
§ 15. The identities (18) to (23) can also be proved by trigonometry in another 
distinct manner, by starting from the trigonometrical sides of the equations. Thus, 
for (18), from the formula 
4 SeC ^ l 7r /3=l2 + ( 32 — 3 2 + |S * + 5 2 + /3 2 — &C-, 
we have (writing z for — for brevity) 
7 sech cos 
4 2p. n* 
3 cos z 5 cos z „ 
+7275 &c. 
3V 
+ 1 
5V 
+ 1 
