MEMBRANES AND PLATES 



145 



If we assume, as is necessarily the case when the origin is 

 included within the region to which (2) applies, that f can be 

 expanded in a series of ascending powers of r, the coefficients 

 (after the first) may be found by substitution in (2), and we 

 obtain 



(3) 



provided 



2 2 .4 2 



.(4) 



This is the Bessel's Function* of zero order, "of the first kind," 

 which we have already met with in 31 ; it is represented 

 graphically in Fig. 51. If a be the radius of the boundary, 



Fig. 51. 



supposed fixed, the admissible values of k and thence of n are 

 determined by the equation 



J (ka) = 0,_ ,...(5) 



viz. we have 



kal-n- = 7655, 17571, 2'7546, 3'7534, (6) 



the numbers tending to the form ra J, where m is integral. 

 The first of these roots corresponds to the gravest of all the 

 normal modes of the membrane. In the rath mode there are 

 in 1 nodal circles, in addition to the edge, whose radii are 

 given by the roots of lower order. Thus in the case of the 

 second root we have for the nodal circle kr/7r = '7655, whence 

 r/a = *4356. The characters of the various normal modes will 

 be understood from Fig. 51, which may be taken to represent 

 a section through the centre, normal to the plane of the 

 membrane. 



* F. w. 



181046. 



I*. 



(17841846), director of the observatory at Konigsberg 



10 



