500 ME. J. W. L. GLAISHEE ON THE THEOEY OF ELLIPTIC FUNCTIONS. 
<px -f <p (x — p) -f <p (x -f (t) -f- &c. = 
then 
fx +f{x - (l) +f(x + ^) + &c.= 
and if 
<px — <p(x—p) — Cp(x + p ) + &c. = 
then 
fo—ftx—p) -f(x+p) + &c. = 
^{/(0)+2/(^)cos^ + 2/(^)cos 4 f +& o. 
^|,( 0 ) + 2 ,(f)co S 2 f + 2 ? (^)cos^ +& c. 
2i/(2 *){n. V \ xr r/3*\ Q ) 
* -{/( ^ / 008 7+A^/ C0S Y” +&c t 
2 \/ (2 ir) ( / 7T \ ttx /SoA Snx 0 ) 
~r~ A cos 7 +*{j ) cos v +&c t 
Also, from the second integral, <p and /'being uneven, if 
<px+<p(x— [A)+<f>(x +(*)-{- See. = 
then 
fx+ftx—yj) +f(x -\-p) +&c. = 
and if 
<px-<p(x—p) — <p(x- j-^)+&c.= 
then 
fx-f(x-p) -f(x+p) + &c. = ■ 
2 \Z(2tt) ( j, / 2%\ . 2 trx /»/ 4.7T \ . 4isx „ * 
—{At) sin ir+/(7) sm ir+ &c '}’ 
■{HD sin f +A?) ,in? f + &c -}- 
2\/( 2tt 
Applying these formulae to the identities (18) to (23), we see that (20) is its own 
reciprocal, as also is the case with (18), (22), and (28); while (19) and (23) are reci- 
procal to one another. Although Cauchy, in his memoir “ Sur les Fonctions reciproques ” 
(Exercices de Mathematiques, seconde annee, 1827), has deduced, by means of his cal- 
culus of residues, a theorem which is in fact (24), he does not appear to have specially 
remarked the reciprocal character of the equations. 
The application of the formulae presents no difficulty. For example, comparing (18) 
with the first of the second pair, we have 
<p#=sech x, 
whence the reciprocal formula is 
fx= 
2 ) .sech 
\J ^jsech sech ^ 7 ^ — sech 
7 TX 
~2 9 
v/ ^ -|sech - cos- + sech — cos^^-|-&c. i, 
P l P P V- V- ) 
which, on replacing \icx and \<7r^ by x and ^ respectively, coincides with the original 
formula (18) 
§ 10. On looking at the formulae (18) to (23) it appears that although we have trans- 
formations for sech x + sech ( x — + sech (#-f-ft<) 4 -&c., cosech x + cosech {x — p) 
