§ 1. The variations of the surface-level of lakes due to the direct action of wind and rain, and the smaller disturbances caused by surface waves, of small or moderate length, due to the action of the wind and the movement of boats and animals, must have been familiar phenomena at all times. The first accurately recorded observation, that lake-levels are subject to a rhythmic variation, similar in some respects to the ocean tides, seems to have been made at Geneva in 1730 by Fatio de Duillier, a well-known Swiss engineer. Owing to the peculiar configuration of the Geneva end of Lake Léman, these variations occasionally reach a magnitude of 5 or even 6 feet; and Duillier mentions that they were known in his time by the local name of “Seiches,” which has now been applied to rhythmic alterations of the level of lakes in general.
If we consider the circle circumscribing any triangle ABO (see figures 11, 12), and diminish its radius still causing it to pass through A and B : then if AOB be an acute angle, C passes without the circle, but if AOB be an obtuse angle, C remains within the circle. If C be a right angle, the radius of the circle, being JAB, cannot be farther diminished.This result, which will be used in what follows, leads in the first place to the result, otherwise obvious, that the minimum circle enclosing three points is the circle through them when they form an acute-angled or right-angled triangle, but the circle having as diameter the line joining the two which are most distant when the triangle which they form is obtuse-angled.Let there be n points. We can always select m of them which form the vertices of a convex polygon enclosing all the others. The problem therefore reduces to finding the minimum circle enclosing a convex »n-gon.Since the m-gon lies wholly on one side of the unlimited straight line, of which any one of its sides is a part, we may regard this line as a circle of infinite radius wholly enclosing the m-gon. Let us diminish the radius of this circle continuously (always supposing it to pass through two particular vertices of the m-gon). Then one or other of two things will happen. Either the radius will diminish down to its minimum value (half the selected side of the m-gon) before the circle passes over any vertex of the m-gon; in which case the circle on the selected side as diameter contains all the n points, and must be the minimum circle, since no less circle can contain the two vertices started with : or the circle will first pass over a third vertex of the m-gon. In the latter case if the three vertices involved form an acute-angled triangle, the minimum circle is reached, for this circle contains all the n points and is the least that can contain the three in question ; if the triangle be obtuse-angled, we observe first that
I propose in this preliminary communication to lay before the Society some results of investigations in the theory of Seiches in a lake whose line of maximum depth is approximately straight, and whose depth, cross section, and surface breadth do not vary rapidly from point to point.As the seiche disturbance is small compared with the length of the lake, I shall make the assumptions usual in the theory of long waves:—viz., that the squares of the displacements and of their derivatives are negligible.
§ 1. In the practical calculation of the periods and nodes of the lakes we have hitherto examined it has been found that the use of the Seiche Functions gives the best results. Indeed, as will appear from details presently to be submitted to the Society by Mr E. M. Wedderburn and myself, the agreement between theory and observation, so far as we have gone, is beyond what might reasonably have been expected. Also the phenomena of concave lakes, i.e. such as have no shallows or points of minimum depth, are easily deducible from the formulæ given in an abstract (Proc. R.S.E., vol. xxv. p. 328, 6th Oct. 1904) which I communicated to the Society on 18th July 1904. On the other hand, the theory of convex lakes is less easy of manipulation, chiefly owing to the difficulty in calculating the roots of the equations
The theory of the singular solutions of differential equations of the first order, even in the interesting and suggestive form due to Professor Cayley (Mess. Math., ii., 1872), as given in English text-books, is defective, inasmuch as it gives no indication as to what are normal and what are abnormal phenomena. Moreover, Cayley added an appendix to his theory regarding the circumstances under which a singular solution exists, which is misleading so far as the theory of differential equations is concerned, if not altogether erroneous.The main purpose of the following notes is to throw light on the point last mentioned by means of a number of examples. I have also taken the opportunity to furnish simple demonstrations of several well-known theorems regarding the p-discriminant which do not find a place in the current English text-books.
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