186-187] CIRCULAR BASIN. 305 



complete primitive consists, of course, of the sum of two definite 

 functions of r, each multiplied by an arbitrary constant, but in the 

 present problem we are restricted to a solution which shall be finite 

 at the origin. This is easily obtained in the form of an ascending 

 series ; thus, in the ordinary notation of BesseFs Functions, we have 



where, on the usual convention as to the numerical factor, 



7" / \ I "1 ^^ / A \ 



~ 2* . * ! t 2 (2s + 2) + 2.4(2* + 2)(2* + 4) ' ' ) ' 

 Hence the various normal modes are given by 



(5), 



smj 



where s may have any of the values 0, 1, 2, 3,..., and A 8 is an 

 arbitrary constant. The admissible values of k are determined by 

 the condition that d/dr = for r = a, or 



J 8 '(ka) = ........................... (6). 



The corresponding 'speeds' (<r) of the oscillations are then given 



The analytical theory of Bessel's Functions is treated more or less fully in 

 various works to which reference is made below*. It appears that for large 

 values of z we may put 



, (! 2 -4s 2 )(3 2 -4s 2 ) 



where J-i-L '+... 



l 2 -4s 2 (I 2 - 4s 2 ) (3 2 - 4s 2 ) (5 2 - 4s 2 ) 

 1.80 1.2. 



.(ii). 



The series P, Q are of the kind known as ' semi -convergent,' i.e. although for 

 large values of z the successive terms may diminish for a time, they ultimately 

 increase again, but if we stop at a small term we get an approximately 

 correct result. 



* Lommel, Studien ueber die Bessel'schen Functionen, Leipzig, 1868 ; Heine, 

 Kugelfunctionen, Arts. 42,..., 57,...; Todhunter, Functions of Laplace, Lame, and 

 Bessel; Byerly, On Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal 

 Harmonics, Boston, U.S.A., 1893; see also Forsyth, Differential Equations, c. v. 

 An ample account of the matter, from the physical point of view, will be found in 

 Lord Rayleigh's Theory of Sound, cc. ix., xviii., with many interesting applications. 



Numerical tables of the functions have been calculated by Bessel, and Hansen, 

 and (more recently) by Meissel (Berl. Abh., 1888). Hansen's tables are reproduced 

 by Lommel, and (partially) by Lord Rayleigh and Byerly. 



L. 20 



