630 PROFESSOR CHRYSTAL 
shifting of the deepest point of the biparabolic lake without alteration of the length or 
maximum depth does not alter the whole volume of water, general dynamical considera- _ 
tions regarding energy would lead us to expect that increase of p would lengthen all 
the seiche periods; and, in point of fact, in the semiparabolic lake, which may be 
regarded as the limiting case of a biparabolic lake when p=, all the periods are greater 
than in the complete parabolic lake, which corresponds to p= 1. 
SEICHES IN AN UNSYMMETRICAL LAKE WITH ONE SHALLOW 
AND TWO MAXIMUM DEPTHS. 
Od DG EB ae 
rea Fe 
ues fic 
§ 40. A good approximation to the form in many cases that occur in nature can be 
obtained by piecing together six parabolee, so as to form a continuous curve. 
Let s be the minimum; and, fh’ the two maximum depths. D and D’ the points 
of inflexion (the depths at which cannot be arbitrarily assigned). Let AB=a,, BD=6, 
DO=d, D’/0’=d’, B/D’=0’, A’B’=a,’; then, for the continuity of the curve of 
longitudinal section at D and D’ we have the following conditions, the laws of depth 
being h(x) =h x (1—a’/a,’) for AB, h(x) =h x (1 —2?/a,”) for BD, A(x) =s x (1 +27/ag*) 
for OD, h(x) =s x (1+2°/a’',”) for OD’, ete. :— 
hbja? — sd/a.2=0 , hb?/a," + sd?/a,2=h—-s; 
h'b'/a’2 —sd'/a'.2=0, h'b?/a',2 + sd?/a'.2=h'—s. 
These lead to 
a? =hb(d+h)/(h—s) , a,” = sd(d + b)/(h — 8) ; (48) 
a2=Nb(d' +0)/(l' =s), a’ 2=sd'(d'+b')/(h—-s). J” 
With the exception of a,, d,, d,, a, and the depths at D and D’, the other 
quantities may be arbitrarily chosen. 
If now 
v,=a/a,, Up = 2/ Ay, U3 = %/ds 5 
, / , / , , 
Dl an Vp =n > eae Gait 
Wy = b/d , Ww, = d/as ; 
0, =0' fa, , Ds =O de; 
c, =n?a,7/gh, c= na,2/gh , Cc, = NA,"/gh ; 
then we have for the various sections A, B, ete. 
Eh(1 =v,”) = {A,C(c,, v1) + B,S(c,, v,) }sin n(t-7), 
C= ZA) + B,S'(c,, v,)}sin n(t—7) , 
etc. 
The origin for x being in each case the vertex of the corresponding parabola. 
