836 
Proceedings of Royal Society of Edinburgh. [sess. 
The values of C and C' which determine the periods are given 
by cjc' = a 2 /a 2 together with the period-equation 
a , C(c,l)S(c',l) + aC(c , ,l)S(c,]) = 0 . . (42) 
If we put a 2 c = a 2 c =n 2 a 2 a 2 /gh = z, the period equation may he 
written 
f 1 - — 1 
1 (\ - z 
V 1.2 a 2 / 
V 3.4a 2 / ‘ 
■ ‘ ■ V 2.3a' 2 / 
' \ 4. 
l - \ 
(\ - z ^ 
( i _ z Vi_ 
i .2a' 2 / 
V 3.4a' 2 / ■ 
’ V 2.3a 2 / \ 
4.5a 2 / 
= 0 . . (43) 
Unsymmetric Lake with one Shallow and two Maximum 
Depths. 
a'ci! 8 l f f ' o d j) /> 2 « /j 
Fig. 4. 
A good approximation to the form of lake section in many cases 
that occur in nature can be obtained by piecing together six 
parabolas, as in figure (4), so as to form one continuous curve. If 
s be the minimum, and h and li the two maximum depths, D and 
D' the points of inflexion ; AB = « 1 , A'B' = a\, BD = 6, B' D' = b\ 
0 D = d, OJ}' = d\ then we may represent the portions A B, B D, 
D 0, 0 D', D' B', B' A' by the six parabolas : — h(cc) = h( 1 - x 2 {a 2 ) ; 
h(x) = h( 1 - x 2 la 2 ) ; h(x) = s(l + x 2 /af) ; h(x) = s(l + x 2 la 2 ) ; li\x) 
= ti( 1 - x 2 la f ) ; h(x) = li\ 1 - x 2 \a!f). 
The conditions of continuity lead to 
a 2 = hb(d + b)/(h - s), a 3 2 = sd(d + b)/(h- s) ; 
a 2 2 = h'b'(d! + £')/(// - s), a'f = sd\d' +- b')/(h' - s) . (44) 
All the magnitudes marked in the figure may be arbitrarily 
determined ; but after this has been done the depths at the points 
of inflexion are not at our disposal. 
The formulae for £ and £ and the period-equation have been 
worked out for this case. They involve all the four seiche- 
