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 sO or sin s0 t 

 the equation (4) takes the form 



This can be integrated by series. Thus, assuming 



m 



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

 between consecutive coefficients is found to be 



(T CL 



or, if we write j 



where n is not as yet assumed to be integral, 



(m*- S *)A m =(m-n)(m + n-2)A m _ 2 ..................... (v). 



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

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

 5 + 4,... in (v). We thus find 



2 (2s + 2) a? 

 or in the usual notation of hypergeometric series 



where a 



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 * + 2;'. 

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



In the symmetrical modes (s = 0) we have 



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

 expression vanishes for j I values of r between and a, indicating the exist- 

 ence of j 1 nodal circles. The value of <r is given by 



l)& .............................. (ix). 



