Purser — Applications of BesseVs Functions to Physics. 55 



Origin at Middle of Axis. 



These expressions give dilatation ^ = 0, and satisfy the internal 

 equations of equilibrium. To calculate the surface forces we have 



du dw ^ my . ^, . . . 



dz dx r 



dv dw ^ . X 



dz dy r 



The corresponding stresses are then given by 



iYi = 2ix-^xy 2m^,»n8(wz) (where ^ = Jonir - '^^Ji{mr)). 

 N, = - N,, = 0. 



X 



Ti = - fji- 'XA,„Ji{mr) (T{mz)m ; 



y 



^2 = ft - %A„jJx{mr) a- mz m ; 

 r 



r 



It appears, then, that the components of stress on element-plane 

 perpendicular to r are given by 



X= yS^^n o-(m), Y = - x^A^^iTKinz), Z=0. 



If, then, the m system be so chosen that = 0, for r = a, it 

 appears that the curved surface is unacted on by stress. 



Consider, now, the forces on terminal planes. These will be 



V X 



2 - mA,n^i{tnr) 8 {tnl), - 2 - mA,„Ji{mr) 8 {ml), 0, 



yielding torque ^mrA,„J^{mr) 8 {ml)^ or shearing force 



^mA„J^{mr)h{ml). 



iS^ow, we can prove that any function of r can be exprcs?;ed 

 between r = 0, r = a, in a series of the form 2a,„/,:^Mr), where 



