1904-5.] Prof. Chrystal on Mathematical Theory of Seiches. 643 
§ 10. For a truly convex lake h cannot vanish, and we have 
$2 _ // 4^- 2 /* 2 - 1 \ _ 1 //1-A 2 /4t 2 \ 
Sj v V167T 2 /* a - 1/ 2 v \l-/£ 2 /167rV’ 
• • • (36) 
which is the characteristic property of convex lakes. 
§11. The particular case where p= — q = a is of some interest. 
We then have k = ir and y = 2. Therefore 
%, = Kit J{gd(w- 1)}, 
= ttI/ J{gd(2v-l)(2v + l); . . (37) 
so that the periods are inversely proportional to the square roots 
of integers as in the case of concave parabolic lakes. 
For the present special case we have 
^ 2 /^ 1 = x/(U/3.5) = J5/5 = *447 . . (38) 
If we put q= —p and r = d we return to the formula for 
a flat lake, as in § 8. 
§ 12. There is no difficulty in calculating the position of the 
nodes of the various seiches by means of the formula (5). But, 
as numerical results are not in immediate view, the length of the 
calculations is too great to justify their insertion in the present 
paper. 
Comparison with the Period Formula of Du Boys. 
§ 13. If we denote the period of the v-nodal seiche, as calculated 
by the formula of Du Boys,* by d T v , we have for a purely concave 
quartic lake 
fT„ = 
dx 
v f(gh)J q a 2 -x 2 
vaj(yh ) log 
j a +p i a + q \ 
( a -pi a - 
2f 
Hence 
T 
d x v _ 
T ~ 
x V 
k 
va Jigh ) 
• (39) 
• (40) 
and, in particular, 
( 41 ) 
“ Essai theorique sur les Seiches Arch. Geneve , xxv. 628, 1891. 
