^{page 599 note *} For the convenience of those who are more interested in the observation of seiches than in the purely mathematical theory, I have separated the mathematics, so far as possible, from the general statement of the conclusions arrived at and the suggestions of further problems to be solved by experiment or observation.

^{page 600 note *} See the extension of Forel's bibliography appended to this paper.

^{page 601 note *} See Airy, art. “Tides and Waves”, §§ 187 et seqq., Encyclopaedia Metropolitana, 1848.

^{page 602 note *} See Proc. Roy. Soc. Edin., vol. xxv. p. 328, Mar. 20, 1905.

^{page 602 note †} Ed. Bull., vol. xl. 149, Feb. 3, 1904.

^{page 603 note *} See § 12 below.

^{page 603 note †} Nevertheless it is curious to pursue this numerical case a little further. Referring to my paper already quoted, and calculating k as above, we get k = 8·112. If we assume the longitudinal section to be symmetrical, then γ = 2 tanh (k/4); and we have γ = 1·932. Hence r = {1 − (γ/2)²}^2d, gives, if we put d = 252^m. (the maximum depth of Constance), r = 1·1^m. If then we take a symmetric quartic lake having the same length as Constance, viz., 65^km, the same maximum depth, and end depths of 1·1^m, we find T₁ = π65 × 10₅/·966√{981 × 25200 × 8÷5} =56′·0. Hence T₁ = 56′·0, T₂ = 38′·4, T₃ = 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.

^{page 603 note ‡} It is interesting to notice that in the case of a concave lake ρ/√∈ is the period of the “anomalous seiche.” See Proc. R.S.E., xxv. (1905), p. 615.

^{page 605 note *} See Forel, Le Léman, 1. ii. p. 148.

^{page 605 note †} Since this was written I have noticed that Endrös, in his able analysis of the seiches of the Chiemsee, cites examples of variations in the phases and amplitudes of nearly pure seiches, which he regards as due to the interference of seiches of the same period differing in phase. He suggests, with great probability, that such seiches are generated by a common but intermittent cause of disturbance.

^{page 606 note *} Dr Endrös has found a striking example in the uninodal seiche of the Waginger See, of which he was good enough to tell me by letter.

^{page 607 note *} It is just possible that this seiche may have been maintained by some continuing but partly intermittent external cause. The limnogram given by Forel seems to show traces of the interference to which Endrös has called attention.

^{page 607 note †} Arch. de Sc. Phys. et Nat. Genève, 3^me Pér. t. iii. p. 1, 1880. The method has been elaborated and used with great effect by Endrös, l.c., p. 15.

^{page 608 note *} This was written before I had access to the monograph of Endrös, whose plan of observation for the Chiemsee is in many ways a model.

^{page 609 note *} There seems to have been a good deal of confusion and some false analogy in this respect.

^{page 610 note *} With a Koenig's fork (256 Ut₂), 140 cm. of No. 30 thread, which weighs ·00058 gm. per cm., and a tension of 45 m., 2·5 mg. of wax produced a displacement of the ventral point exceeding 8 cm.

^{page 611 note *} See Proc. R.S.E., vol. xxv. p. 646; also § 49 below.

^{page 611 note †} Trans. Roy. Soc. Edin., xiv. p. 524, 1839.

^{page 612 note *} To incline the trough, keeping the volume of water the same, is not the same thing.

^{page 612 note †} Endrös commenced his investigation of the complicated seiche-phenomena in the Chiemsee with a series of experiments on the oscillation of mercury in a vessel imitating the configuration of the lake, the results of which, on the whole, were in remarkable agreement with his subsequent observations, l.c., p. 8.

^{page 613 note *} In order that these assumptions may be justified, the square of the ratio of the depth to the wave length must be negligible at every part of the lake. See Lamb's Hydrodynamics (1895), § 169.

^{page 615 note *} See Rayleigh's Sound (1877), vol. i. § 142.

^{page 619 note *} See Whittaker's Modern Analysis (1902), § 96.

^{page 621 note *} Since the abstract of this paper was published, I have discovered that the solulion for the particular case of a symmetric parabolic basin was given by Lamb in the new edition of his Hydrodynamics (1895), § 182. He arrives at his result by means of Legendre's function, which is closely allied to the seiche functions.

^{page 629 note *} In the general case h is the maximum value of the product of the area of a cross section by its surface breadth; and a and a′ the areas of the lake surface between the corresponding section and the ends.

^{page 636 note *} Readers unacquainted with the properties of Bessel Functions will find all that is here required in a few rices of the treatise by Gray and Mathews (1895), ch. ii., and pp. 241–292 containing the tables.

^{page 637 note *} Roots of the Bessel Functions.—In what follows I shall denote the positive roots of the equation J₀(z)=0 by j ₁, j ₃, j ₆ . . . . .; and the positive roots of J₁ (z)=0 (excluding the zero root j ₀ = 0) by j ₂, j ₄, j ₆, . . . . . . . . So that we have approximately j ₁ = 2·405, j2 = 3·832, j ₃ = 5·520, j ₄ = 7·016, j ₅ = 8·654, j ₆ = 10·173, j ₇ = 11·792, j ₈ = 13·323, j ₉ = 14·931, j ₁₀ = 16·471, etc.

For large values of n, j_n = (2n + 1)π/4, approximately: e.g. this formula gives j _n = 18·064 instead of the correct value 18·071; so that the error after n = 11 is less than ·1 %.

^{page 638 note *} These formulæ are given by Lamb in his Hydrodynamics (1895), § 182.