Vibrations in Solid and Hollow Cylinders. 343 



The displacements are u outwards along r, the perpendi- 

 cular on the axis, and w parallel to the axis, taken as that of 

 z. Thomson and Tait's notation 711, n for the elastic constants 

 is employed. TJie frequency is k/2w and the density p, as in 

 the earlier part of this paper, and for brevity 



&p/(m + n)=J, k 9 p/n = P, . . . (25) 



so that a. and ft have utterly different significations from 

 their previous ones. 



There being no displacement perpendicular to r, in a trans- 

 verse section, the dilatation 8 is given by 



5. du u dio , x 



S= A- + r + dz ^ 



It was shown in paper (A) that the following equations held 

 #8 .ldB. d*8 



d? + v<h- + d;* +aB=0 > • • • • ( 2? ) 



*£ + !£_« +£+,<*,= _*<» (28 ) 

 dr* r dr r 2 dz l n dr K ' 



d 2 w , 1 div , d 2 w -g m d8 ,_. 



f/r* 2 ?' dr a* 2 rc dz 



Employing J and Y to represent the two solutions of 

 BessePs equation we find, as in paper (A), that the above 

 equations are satisfied by 



8 = cos kt cos (pz - e) { G J {r (a 2 - f)h) + 0' Y (K" 2 -p 2 )h) } , (30) 

 11= cos ft cos {pz-e) [AJ\{r(/3 2 -p 2 )i) + A'Y l (r{/3 2 -p^) 



~f i iE^\ GJ M^P 2 ) i ) +O r Y 1 (r(««-^*)}], (31) 

 mj=- cos fa sin Cp*-e) R AJ (?'(/3 2 -^ 2 )*) + A'Y (r(/3 2 -p 2 )t) j- 

 x (fag + ^j*^ {OJo(Ka»-^)») + PYo(r(a»-^)i)}] > (32) 



where A, A', C, C are arbitrary constants to be determined 

 by the surface conditions. 



In reality ofi-p 2 is negative ; but the properties of the 

 J and Y Bessel functions which at present concern us 

 are not affected thereby. ft 2 —p 2 , on the other hand, is 

 positive. 



