638 Proceedings of Royal Society of Edinburgh. [sess. 
§ 4. Since our object at present is general explanation, nothing 
will be lost by supposing the lake to be of uniform breadth and 
rectangular cross-section. In that case, using the notation of the 
abstract above referred to, we shall have 
h(x)£ — u = P sin nt ; 
du 
£ = " Tx’ • 
(*) 
(5) 
where £ and £ are the horizontal and vertical displacements of the 
seiche ; and h(x) = h(a 2 + x 2 ) 2 is the depth at a distance x from the 
origin.* 
The differential equation for P is now 
<P P cP 
dx 2 + (a 2 + x 2 ) 2 
where 
c = n 2 !gh . 
( 6 ) 
(7) 
Concave Lakes. 
5. The solution of the appropriate differential equation, viz. — 
d 2 P cP _ 0 
dx* + (a a -a?) s 
is P = A {a + x) m (a - x) n + B(a + x) n (a - x) m , . 
m and n being the roots of 
p 2 - p + c/^a 2 = 0 ; 
Say m — 1/2 + di , n = 1/2 — di , where c = a 2 ( 4d 2 + 1) . . 
We thus get 
P . L. -^{ i 
= (a 2 - x 2 Y i C cos ( 0 log a ^ X \ + D sin ( d log a x \ 1 
I \ a - xj \ a - xj J 
(8) 
(9) 
( 10 ) 
( 11 ) 
( 12 ) 
C and D being arbitrary constants. 
The longitudinal section of such a lake as we are now 
discussing is indicated by the upper curve in the figure on 
p. 639. As we approach very near the infinitely shallow ends 
* When breadth of the lake is not uniform, and the form of the cross- 
section varies, we must use the normal curve ; but in practice this merely 
requires an alteration in the meaning of the constants h and a. 
