Secondary Undulations of Oceanic Tides. 109 
length of the lake and also uniform in each section. The 
vertical acceleration is neglected in comparison with the 
horizontal. 
Take the origin of rectangular coordinates at one [endiof 
the lake, x axis being in the direction of length, f, 77, b have 
the same meaning as before. Then the kinetic and the 
potential energy are given by 
K.E. = ij/aSf^ and P.E. = ^gbp V 2 d^ 
respectively. Again, from the condition of continuity, 
where X= Sf . By the condition of the constancy of energy, 
we have 
r?^ + 4»'K§) 3&=c ° nst ' 
Assuming for the first approximation 
X.=a sin-y- cos nt, 
substituting this value in the above equality, and putting 
T 
£ = and t= T , two relations are obtained ; whence the ex- 
4: 
pression for the period of oscillation follows at once : — 
2! f.lf 1 2irx.bJ> ASo _ 
L= v^i 1+ 4 C0S ^^ + ^))' • • (3) 
where lb = surface-area and 7* /> = S . Here the expression 
may be considered as the correction to the length. It shows 
that any contraction or expansion towards the central part 
of the lake prolongs or shortens its natural period respec- 
tively, and that a contraction or expansion towards the ends 
shortens or prolongs it respectively. 
To apply the above expression in the case of a bay, we 
need only to consider a lake whose shape is symmetrical 
with respect to the vertical plane through the mouth line, 
and to find the period of the seiches in the lake by the above 
formula. This period, if it be corrected for the mouth, is 
the required period of oscillation in the bay. 
