188-189] BASIN OF VARIABLE DEPTH. 313 



As a simple example we may take the case of a circular basin which 

 shelves gradually from the centre to the edge, according to the law 



Introducing polar coordinates, and assuming that varies as cos sQ or sin sO, 

 the equation (4) takes the form 



This can be integrated by series. Thus, assuming 



,m 



LJ 



where the trigonometrical factors are omitted, for shortness, the relation 

 between consecutive coefficients is found to be 



= m (m - 2) - ^ - 



L m 2&amp;gt; 



or, if we write Tr-*n(n 2) ^ (iv), 



where 7& is not as yet assumed to be integral, 



The equation is therefore satisfied by a series of the form (iii), beginning with 

 the term A a (r/a) 8 , the succeeding coefficients being determined by putting 

 m=5 + 2, s + 4,... in (v). We thus find 



(m-t- 



8 \al I 2(&? + 2) a z 





or in the usual notation of hypergeometric series 



where a = -^7i + is, /3 



Since these make y a /3 = 0, the series is not convergent for r = a, unless it 

 terminate. This can only happen when n is integral, of the form 

 The corresponding values of o- are then given by (iv). 



In the symmetrical modes (5=0) we have 



where j may be any integer greater than unity. It may be shewn that this 

 expression vanishes for^ - 1 values of r between and a, indicating the exist 

 ence of j - I nodal circles. The value of or is given by 



() 





