ON THE HYDRODYNAMICAL THEORY OF SEICHES. 603 
behaves not very differently from a quartic lake of the kind supposed. Of course it by 
no means follows that the normal curve* of the Lake of Constance is even 
approximately like the quartic curve supposed.¢ The point I wish to make is that, 
from the hydrodynamical point of view, there is nothing surprising in the relation of 
its periods. Farther investigation of the phases of the seiche at different parts of the 
lake will probably show that the seiche a la .quinte, as Foret calls it, for which 
T=39’, is really the binodal seiche; and that the seiche for which T=28'1’ is a 
trinodal. 
§ 8. The formula for the periods of a quartic lake given above may be written 
T,=p//(’ +e), where p=hl/y,/(gd), «=+k'/47’, the minus sign corresponding to 
convex lakes. As this formula applies to quartic lakes of every variety, concave or 
convex, symmetric or asymmetric, it may be conjectured that it will give rough 
approximations at least to the periods of any actual lake of fairly regular configuration. 
The constants could not be determined w priori without discussing the bathymetric 
data for each lake. The ratios of the periods, however, depend only on the single con- 
stant e. We have, in fact, T,/T, = ./{(1+¢)/(’+.¢)}. In general the two longest periods 
are best known. If we take these as given, the equation T,’/T,’=(1+¢)/(4+¢) gives 
the value of «; and then T,=T,,/{(1+¢)/(’+6)} =p/ /(’ +e), where p=T,,./(1+¢).f 
For very large values of v, we have approximately T,=p/v; in other words, the periods 
of the seiches of higher nodality approximate to a harmonic series. By means of these 
formulee—which I shall call the Quartic approximation—the table on p. 604 has been 
calculated for a number of lakes whose periods are fairly well determined. _I have 
gone in most cases as far as T=4°7’, which is the period found by Forer for the 
longest progressive surface waves ever observed on Léman, calculated for infinite depth 
of water. 
As a control, I have added at the end of the table, under A, B, C, D, the periods for 
a complete symmetric rectilinear, semirectilinear, complete parabolic, and semiparabolic 
lake respectively, as calculated by the quartic approximation, and as calculated by the 
accurate formulee of §§ 27, 34, 49, and 51 below. 
* See § 12 below. 
+ Nevertheless it is curious to pursue this numerical case a little further. Referring to my paper already quoted, 
and calculating & as above, we get k=8'112. If we assume the longitudinal section to be symmetrical, then y=2 tanh 
(k/4); and we have y=1:932. Hence r= {1-(7/2)?}?d, gives, if we put d=252™. (the maximum depth of Constance), 
7=11™ If then we take a symmetric quartic lake having the same length as Constance, viz., 65*™., the same 
maximum depth, and end depths of 1:1™, we find T, = 165 x 105/:966,/{981 x 25200 x 8+5}=56"0, Hence T,=56"0), 
T,=38"4, T,=28'. The agreement with the observed periods of Constance is curiously close, and is, no doubt, 
partly accidental. It will be of great interest to work out the normal curve for Constance, and calculate the periods 
by a rigorous application of the theory, as has been done by Mr WEDDERBURN and myself for Treig and Earn. 
f It is interesting to notice that in the case of a concave lake p/,/« is the period of the “anomalous seiche.” See 
Proc, R.S.E., xxv. (1905), p. 645. 
