640 Proceedings of Royal Society of Edinburgh. [sess. 
we must have 
C cos 0a + D sin 0a = 0 ; C cos d/3 + D sin d/3 = 0 . 
Since C and D cannot both vanish, these lead to 
cos 0a sin d/3 — sin da cos dj8 — 0 ; 
that is, 
sin(a-/3)0 = 0. . . . (14) 
Excluding the case 0 = 0, to which we shall return later on, 
(14) gives 
0 = v7r/(a - fi) = V7 r/k , say, . . . (15) 
where 
Hence, for the v-nodal seiche, 
= a 2 (40 2 + 1) = a 2 (4v 2 7r 2 /F+ 1) ; . . . (17) 
T„ 4 L jn = 2tt /a J{gh( ivV/p+l)}. . (18) 
§ 6. lid be the maximum depth, corresponding to £ = 0, and 
r and s the depths corresponding to and x = q respectively, 
we have d = r = h(a 2 - p 2 ) 2 , s = h(a 2 - q 2 ) 2 ; and, if l be the 
length of the lake, L = p- q . Therefore 
V( 1_ Vs)’ ? = *%/(* ■ (19) 
I “ a {\/( 1 '\/a) ± \/( 1 “\/ )0} = ar ’ say - (20) 
Hence the period of the v-nodal seiche is given by . 
where 
r = 
and 
i + 
k = loi 
T, = 2ir//y J{gd(4:V 2 TT 2 /k 2 + I )} ; 
l + s/( 1 -y/d) l 1 + \Z( U ' \/a)l 
• ( 21 ) 
l -y/( l ~V r d) l± y/{ l ~Vi)\' 
(23) 
In the particular case of a symmetric lake, q= - p , fS ; and 
(22) and (23) take the simpler forms 
y -Vo \A)\ • • ■ (22,) 
k = 21o s{j j ~)- ■ ■ 
(23') 
