ME. J. W. L. GrLAISHEE ON THE THEOET OE ELLIPTIC FUNCTIONS. 
493 
It must be remarked that in (11) and (14) the number of terms must always be 
uneven; this point will be noticed at greater length further on (§ 10). 
§ 5. Writing the hyperbolic sine, cosine, &c. as sinh, cosh, &c., these formulae may 
also be written in a somewhat different form : thus 
cos am w=5£j£/j sech sech ^(w— 2K)— sech ^7 (w+2K)+&c.j, 
sin am M=^y|tanh tanh (u— 2K) — tanh gjjy (w+2K) + &c.j, 
and similarly for the others. 
I do not think it likely that the formulae (10) to (17) are new, but I have not 
succeeded in finding them anywhere. Schellbach (‘ Die Lehre von den elliptisehen 
Integralen . . . ’ Berlin, 1864, p. 38) gives the corresponding forms for du, 6 l u , &c., 
but he does not allude to the similar expressions for the elliptic functions. It would, 
however, in any case have been necessary for the explanation of the rest of this paper 
to have written down the latter and demonstrated one of them. 
§ 6. By equating the values of sin am u, cos am u, &c., as given by (1) to (8) and by 
(10) to (17), we obtain a series of identities of an algebraical character ( i . e. which are 
independent of the notation of elliptic functions). Thus from (2) and (10) we have 
(remembering the definitions of v, &c. at the end of § 3) 
kK\ 
cos x cos 3x cos 5x 
cos 3x cos 5x 
-+■ 
cosh % cosh ^ cosh 
&c.|=^^{sech 2 : — sech(s— v) — sech( 2 ; + v)+ &c.}, 
" ~2 
+ &c. = ^ jsech 7l ~ — sech ^ (# — 7r) — sech ^ (x-\-r) -f- &c.|. 
This may be written (by interchanging x and z, and v) in the rather more conve- 
nient form 
sech x — sech (x— p) — sech (x + sech (x—2 p) -f- sech (x-\- 2p) — &c. 
icx 3nx 5irx 
n , cos — cos cos — 
* f M3 ,^ + oosh^ + cosh^ + ) 
2{X. 2 p p 
In the same way, by comparing (1) and (11), we find 
. % 
, sin 
tanh x — tanh {x— p) — tanh (x-\-(a)-\-&c.=—< -j ~r 2 -\-Scc. > ; 
^ 1 sinh ~ sinh — — ' 
. tcx . 3 nx 
sin — sin 
and by comparing (3) and (12), 
2|* 
sech^-J-sech {x— ^)-j-sech (^+ ( 'A)-f&c.=--jl-{- 
2irx 4%x 
2 cos 2 cos 
f 4 _L 
: o +&C. 
cosh — cosh 
(18) 
(19) 
( 20 ) 
