72 
Proceedings of the Eojal Society of Edinburgh. [Sess. 
Writing for snx, for sns, etc., this becomes 
/0UY 
puy 
V^/ 
+ cosh^ y^{s^d^c<^ - s^d^Jc^) 
+ ^'2 sinh^ i]{c^d-^si^ - c^d.,^s-^) 
+ cosh y] c^d-^c^dj^s-^ - 
H- 2A:^ k' sinh 77 s^d-^s^d^{s-^ - .s^^) 
+ k' cosh 77 sinh 77 ~ 
= ^ 2 /, 2 _ 2 ) pi - A : 2 ( 9,2 + ;/) + ^4 ,^ 2,^2 _ /,'2 ,^2 ,^2 
+ B COsh2 77 { 1 - ( 9^2 + + 9,2 ^’^2 + /^'2 9,2 } 
+ 2^ cosh 77 c-^d-^c^d^ + 2 k k' sinh 77 s-^d-^s^d^ 
+ 2^2 Qosl, yj ginh 77 9,C,92Cg_ 
= U 2 P ) 
With the help of the two properties (6) and (7) it is an easy matter to 
establish the result (2). In fact, if we write 
1 = 
ds 
6^.2 
r^iS. 
U«2/(s) 
J 0 
odic soluti 
= [n{n + \)k‘^ sn^ s + A]y 
where y{s) denotes a doubly-periodic solution of the equation 
d‘^y 
(8) 
we have 
^21 
dx^ 
'4K 
=ri 
>i{n + 1)F 9?z2ir + A}I 
«-2/9U\2 
dx 
Using (6) and (7), the right-hand side becomes 
n(n + 1 )Zj 2 s ^2 xV'^ ~ AU* 
wU” M + 2k‘^{sn^ X - sn^ s)J] +n{n-\)\'J 
n{n + 1)^2 g ^2 X U'^ — AU”" | y(s) ds. 
') + k?{sn^ X - 97^2 s)XJ2 I 
ds J 
y{s) ds, 
or 
ri"”' 
-10^U , , 1MT..-2/3UV 
0?+”(”-i)U" w 
- n{n + 1)7:2 
or by (8) 
f{ 
ds^ j 
or 
kr 
ds J’ 
(79, 
which is zero, since and y{s) have the period 4K. 
