1904-5.] Prof. Chrystal on Mathematical Theory of Seiches. 641 
§ 7. It will be observed that, so long as k does not vanish, which 
cannot happen for a wholly concave lake, we have 
T 2 _ /( ±7rW + 1 \ 1 // 1 + * 2 /4tT 2 \ 
T x V \16ir 2 /& 2 + 1/ 2 V \1 + & 2 / 16 ttV ’ 
(2^) 
which is now a well-established property of concave lakes. 
The case of a flat lake, i.e. a lake of uniform depth, yve have 
r = d, y = 0, k = 0 . Hence, since 
L*/y = L(2/ 7 )Iog{(2 + 7 )/(2-y)} = 2, 
y=0 y — 0 
we have 
Tv = 27rllJ{gdvW} = 2l/vJ(gd), 
as it ought to be. 
In this case, of course, T 2 /T 1 = 1/2 . 
Convex Lakes. 
§ 8. The differential equation for P is now 
0 , 
d 2 P cP 
+ 
dx 2 (a 2 + x 2 ) 2 
where c = n 2 lgh . 
The solution is 
P = A(x - aiff x + aif + B(a? - ai) n (x + ai ) m , 
m and n being the roots of the quadratic 
p 2 - p- c/ia 2 = 0 ; 
say m— 1/2 + 0, n = 1/2 -3 , where c = a 2 ( 43 2 - 1) . 
Hence 
P = (x 2 + a 2 f\ A 
— ai\ 8 
x — ai 
; + b 
x-ai 
0 " 3 }’ 
w J 
(25) 
+ ai) \x + aij 
= (x 2 + a 2 f{ A exp 3(log {x — ai) — log (x + ai) ) 
+ B exp 0( - log (x - ai) + log (x 4- ai ) ) } . 
Therefore, since 
log ( x - ai) = log (x 2 + a 2 f + i tan ^(x/a ) , 
we have 
P = (x 2 + a 2 f | A exp (2id tan" 1 - ^ + B exp ^ - 2 id tan -1 — ^ | 
= ( x 2 + a 2 f | C cos ^23 tan -1 - ^ + D sin^23 tan -1 — ^ j (26) 
PROC. ROY. SOC. EDIN. — YOL. XXY. 
41 
