146 DYNAMICAL THEOEY OF SOUND 



The complete solution of the differential equation (2), 

 which is of the second order, would consist of the sum of 

 two definite functions of kr, each multiplied by an arbitrary 

 constant; but the second solution, which is called a Bessel's 

 Function " of the second kind," becomes infinite for r = 0, and 

 is therefore inapplicable to a complete circular area. In the 

 case of an annular membrane, however, bounded by concentric 

 circles, both solutions would be admissible, and both would be 

 required in order to satisfy the conditions at the two edges*. 



The theory of the symmetrical vibrations of a circular 

 membrane was given by Poisson (1829), who also calculated 

 approximately a few of the roots of the period-equation (5). 



When the vibrations are not symmetrical we may begin by 

 calculating the forces on a quasi-rectangular element of area 

 bounded by two radii vectores and two concentric circles, the 

 sides being accordingly $r and rB0. The stresses on the 

 curved sides give a resultant 



normal to the plane, whilst the stresses on the straight sides 

 produce 



Equating the sum of these expressions to pr$0Sr . , we obtain 



p^pJil^U- 8 ^ (7) 



p tf (r9rV 8rJ r 2 90 2 j' 



or, in the case of simple-harmonic vibrations, 



with the same meaning of & 2 as before. 



* On account of the frequent occurrence of the Bessel's Functions in 

 mathematical physics, especially in two-dimensional problems, great attention 

 has been devoted to them by mathematicians. The difficulty in investigating 

 their properties is much as if we had to ascertain all the properties of the 

 cosine-function from the series 



and were ignorant of its connection with the circle. 



