ON THE HYDRODYNAMICAL THEORY OF SEICHES. 609 
MactiaGan-WEDDERBURN and myself, presently to be submitted to the Society, good 
results for Lochs Earn and Treig have been obtained by using two parabolas. 
§ 16. In the course of the mathematical investigation, an abstract of which was 
published in the Proceedings of the Society in October last, I have had occasion to 
introduce certain new functions, which | have called the Seiche-cosine, Seiche-sine, 
Seiche-cotangent, and the Lake-function. My reason for venturing to give special 
names to these functions is, that I believe that some of them, especially the Seiche- 
cosine and the Seiche-sine, which are independent synectic integrals of a special case 
of the hyper-geometric equation, will be found interesting theoretically, and very 
useful in applications. Preoccupation with the physical side of the present problem 
has prevented me from following the pure mathematics which it suggests. Happily, 
this part of the question has been taken up by a younger mathematician, Dr Ha.m, 
who, in a paper which will be submitted to the Society along with the present 
communication, has obtained a number of interesting results, which tend to justify my 
belief that the seiche functions are not unlikely to occupy a permanent place in applied 
mathematics. He has also given tables for calculating C(c, 1) and S(c, 1), and graphs 
of these functions. 
§ 17. I have said that the acoustic analogy with an organ pipe is unsatisfactory in 
the general case. There is, however, an acoustic analogy which is perfect. Consider 
a stretched string, fixed at both ends, whose length in conveniently chosen units 
represents the median line of the lake, which we take to be of uniform breadth 
and rectangular cross section, but of variable depth, h(x), at any distance « along the 
median line. Suppose the string so constructed that its density at a is 1/h(x); and let 
u=P(x)sinnt be the transverse displacement in any fundamental (normal) mode of 
vibration at time ¢ of the point which was originally at x. Then, if the tension of the 
string be properly adjusted, we shall have 
gh(z) =u, = —du/jdz, 
€ and ¢ denoting the horizontal and vertical displacements of the seiche. The motion 
of the string, therefore, exactly represents the seiche movement, the transverse dis- 
placement of the string corresponding to the horizontal displacement of the seiche ; 
and the gradient of the curve formed by the string at any moment to the vertical dis- 
placement of the seiche. 
It will be noticed that nodes of the string correspond to ventral points of the 
lake ;* and the ventral points on the string to nodes on the lake. This analogy is most 
useful, both in explanation and in suggesting methods of calculation. To a uniform 
string with its binodal, trinodal, quadrinodal, etc. modes of vibration, corresponds a 
lake of uniform depth with its biventral, triventral, quadriventral, etc., 7.e. uninodal, 
binodal, trinodal, quadrinodal, ete. seiches. 
Starting with a uniform string, let us increase its density by adding a small 
* There seems to have been a good deal of confusion and some false analogy in this respect. 
