1G2 164.] WAVES IN TWO DIMENSIONS. 197 



where A n is an arbitrary constant, and J H (kr) denotes the Bessel s 

 Function of the nth order of the variable kr, viz. * 



+ 2) + 2. 4 (2n + 2) (2 + 4) &quot;&quot; &C J 



The summation in (43) is supposed to include all admissible values 

 of k t to be determined by the equation 



_ p* = 0, when r = a, 

 ar 



or J n (ka)=0 (44), 



the accent denoting the first derived function, and a being the 

 radius of the tank. It may be shewn that the values of k satisfy 

 ing (44) are infinite in number and all real. For the particular 

 case n = 0, when the motion is symmetrical about the centre, the 

 lowest roots are given by 



= 1-2197, 2-2330, 3 2383, &c. 



7T 



For a discussion of the various kinds of motion represented by 

 the above formulae the reader is referred to Lord Rayleigh s paper 

 on Waves already citedf, which contains besides a comparison 

 with theory of some experimental measurements of the periods of 

 oscillation in rectangular and circular tanks made by Guthriej. 



Free oscillations of an Ocean of uniform Depth. 



164. We close this chapter with the discussion of the follow 

 ing problem, which is of some interest in connexion with the 

 theory of the tides : To determine the free oscillations of an ocean 

 of uniform depth completely enveloping a spherical earth. The 

 method employed is due to Thomson . 



Let r denote the distance of any point from the centre of the 

 sphere, and a, b the values of r at the surface of the solid sphere, 

 and at the mean level of the ocean, respectively. The general 



* Todhunter, -Functions of Laplace, &c., Art. 370. 



t See also Theory of Sound, c. 9, by the same author. The values of the roots 

 of (44) for the case n = are taken from this source. 



* Phil. Mag. 1875. 



Phil. Trans. 1863, p. 608. 



