310 TIDAL WAVES. [CHAP. VIII 



sections of the free surface by planes through the axis of z is given 

 by the curve y J l (x) on p. 306. 



The first of the two modes here figured has the longest period 

 of all the normal types. In it, the water sways from side to side, 

 much as in the slowest mode of a canal closed at both ends 

 (Art. 175). In the second mode there is a nodal circle, whose 

 radius is given by the lowest root of Jj (kr) = ; this makes 



A comparison of the preceding investigation with the general theory of 

 small oscillations referred to in Art. 165 leads to several important properties 

 of Bessel s Functions. 



In the first place, since the total mass of water is unaltered, we must have 



/; 



where has any one of the forms given by (5). For s&amp;gt;0 this is satisfied in 

 virtue of the trigonometrical factor cos sQ or sin s6 ; in the symmetrical case 

 it gives 



tn, 



(iv). 



r 

 / 



Again, since the most general free motion of the system can be obtained 

 by superposition of the normal modes, each with an arbitrary amplitude and 

 epoch, it follows that any value whatever of , which is subject to the 

 condition (iii), can be expanded in a series of the form 



r) ..................... (v), 



where the summations embrace all integral values of s (including 0) and, for 

 each value of s, all the roots k of (6). If the coefficients A s , B s be regarded 

 as functions of , the equation (v) rnay be regarded as giving the value of the 

 surface-elevation at any instant. The quantities A 8 , B s are then the normal 

 coordinates of the present system (Art. 165) ; and in terms of them the 

 formulae for the kinetic and potential energies must reduce to sums of 

 squares. Taking, for example, the potential energy 



* The oscillations of a liquid in a circular basin of any uniform depth were 

 discussed by Poissoii, &quot; Sur les petites oscillations de 1 eau contenue dans un 

 cylindre,&quot; Ann. de Gergonne, t. xix. p. 225 (1828-9); the theory of Bessel s 

 Functions had not at that date been worked out, and the results were consequently 

 not interpreted. The full solution of the problem, with numerical details, was 

 given independently by Lord Eayleigh, Phil. Mag., April, 1876. 



The investigation in the text is limited, of course, to the case of a depth small 

 in comparison with the radius a. Poisson s and Lord Kayleigh s solution for the 

 case of finite depth will be noticed in the proper place in Chap. ix. 



