130 Dr Glaisher, On expressions for the [Nov. 14 
_ 2 = /[* {sinh 2¢ +sin 2 (¢ +2) 
~ J/arpJo (cosh 2¢+ cos 2 (t+ «) 
sinh 2¢+ sin 2 (t—2) it (=) a 
cosh 2¢ + cos 2(¢ — 2) es i 
e * (sinh ¢sin (¢ + #) — cosht cos (¢+ 2) 
se) Fels cosh 2¢ + cos 2 (¢ + a) 
sinh¢ sin (t — 2) —coshtcos(t—«)) . = 
a cosh 2¢ + cos 2 (¢ — x) ae ee au 
_ 4 [*(sinh tsin (¢+ a) + cosh ¢ cos (¢ + 2) 
orp Jo cosh 2¢ + cos 2 (+ 2) 
_ sinh tsin (¢— #) + cosh ¢ cos (2) , Pe =) Ze 
cosh 2¢ + cos 2 (¢ — x) cos : 
@ _ 4 f° (sinh ¢cos (t+ a) + cosh ¢sin (+ 2) 
= FI, cosh 2¢ — cos 2 (¢ + a) 
sinhtcos (¢—#)+coshtsin(t—a)) . /2° 
cosh 2¢ — cos 2 (t — 2) 7 @ ae 
+ sinh ¢ cos (t+ «) — cosh ¢ sin (¢+ 2) 
sol cosh 2¢— cos 2 (¢+2) 
e sinh ¢ cos (t — x) — cosh ¢ sin (¢ — eo | dt 
cosh 2¢— cos 2 (¢— 2) os (= 3 
@, 4 [° (sinh 2¢ +4 sin 2 (¢+ 2) 
Os Ge cosh 2¢ — cos 2 (¢ + #) 
sinh 2¢+sin2(t—#)) . /2? & 
cosh 2¢ — cos 2 (¢— 2) at & 
_ 4 ff (sinh 2é—sm2(¢-+ 2) 
=F=| cosh 2¢ — cos 2 (¢ + 2) 
sinh 2¢—sin 2 (¢— «)) os (= Ap 
cosh 2¢ — cos 2(f—«){ “° ) 
+ 
These expressions put in evidence the periodicity of the 
functions with respect to 27, the argument x entering into them 
merely in the form of a quantity added to, or subtracted from, the 
variable ¢ in the circular functions. 
