602 PROFESSOR CHRYSTAL 
is given by #,=//2; the binodes by #,=//4, x,=381/4; the trinodes by x, =1/6, 
i, — 31/6, %, — 61/6 ;, and so on; 
In this case the wave length for each pure seiche is the same at all parts of the 
lake; the ventral points are midway between the nodes; and the uninode, middle 
trinode, middle quinquinode, etc. are all at the middle of the Jake. In fact, the 
periods, nodes, and ventral points follow the same law as the periods, nodes, and ventral 
points of the fundamental and over tones of an organ pipe open at both ends. 
§ 4. When, however, the depth of the lake varies, this acoustic analogy is in some 
important respects misleading. The theory of long waves applied to a longitudinal 
seiche in a lake of uniform breadth and rectangular cross section, but varying depth, 
leads to the following among other general results. 
§ 5. In any given lake, seiches of all degrees of nodality, z.e. uninodal, binodal, 
trinodal, ete., are possible; and any actual seiche is either one of these or a super- 
position of several of them. Perhaps the most commonly occurring case is what ForuL 
calls a dicrote seiche, whose components are uninodal and binodal. 
§ 6. The periods of the series of pure seiches are not in general proportional to the 
terms of the harmonic series1,3,4,4,.... The ratios of the periods are in 
general incommensurable; in general, not even algebraic numbers, although in certain 
special cases the periods are inversely proportional to the square roots of integral 
numbers. Thus, for a lake of symmetric complete parabolic longitudinal section, we 
have T,=7l/,/{»(v+1)gh}; so that the ratios are 
TE eT, ae os, = (Lee (cai I oe sae 
Indeed, it follows from a result* which I obtained for lakes whose longitudinal 
section is part of the quartic curve z= h(1 — a?/a2)" that concave lakes can be imagined 
in which T,, T,,T,, ... . all approach as nearly to equality as we please. Hence, 
for example, it may very well happen that it is the trinodal, and not the binodal seiche 
whose period is half the period of the uninodal,—a result wholly in contradiction with 
the acoustic analogy suggested by the consideration of lakes of uniform depth. 
§ 7. As this is a matter which seems to have caused some perplexity, it may 
be well to give some numerical illustrations. Let us take a quartic lake of the kind 
discussed in the paper to which I have referred. The period of the »-nodal seiche 
is given by T, = 2al/y,/{gd(4:*0"/k’? +.1)} , where y and & depend on the configuration 
of the lake. Suppose that T,/T,=1/2, then we find that we must have 47*/k’ = 3/5. 
It then follows that T,/T,=°686. For a lake of this kind we should therefore have 
T,:T,:T,;=1:°686:°5. In a communication to the Société Vaudoise des Sciences 
Naturelles, Forrn{ mentions that three periods have been determined from observa- 
tions on the Lake of Constance, viz., T,=55°8, T,=39°:1, T,=28°1, which give 
T,: T,:T,=1:°701:°'504. It would appear, therefore, that the Lake of Constance 
* See Proc. Roy. Soc, Edin., vol. xxv. p. 328, Mar. 20, 1905. 
+ Hat. Bull. vol. xl. 149, Feb, 3, 1904. 
