134 DYNAMICAL THEORY OF SOUND 



dealt with imperfectly in this book, owing to the difficulties of 

 an exact investigation. As various points of interest arise, we 

 treat the matter somewhat fully. 



The ring is supposed to be uniform, and the section is 

 assumed to be symmetrical with respect to a plane perpen- 

 dicular to the axis. We further consider only vibrations 

 parallel to this plane. Let u, v be the displacements of an 

 element of the ring along and at right angles to the original 

 radius vector, so that the polar coordinates of the element are 

 changed from (a, 0) to (a + u, 6 +- v/a). We require expressions 

 for the extension, and for the change of curvature. In con- 

 sequence of the assumed smallness of the displacements, we 

 may calculate the instalments of these quantities which are due 

 to u and v separately, and add the results. The radial displace- 

 ment by itself changes the length of an element from a&O to 

 (a + u) 80, and so causes an extension u/a. The transverse 

 displacement obviously contributes dv/adO. The total extension 

 is therefore 



.(1) 



Again, in consequence of the radial displacement alone the 

 normal to the curve is rotated backwards so as to make an 

 angle du/adO with the radius, and the mutual inclination of the 

 normals at the ends of an element a BO is accordingly diminished 

 by fru/adfr . S0. Dividing the angle between the normals by 

 the altered length (a + u) BO we get the altered curvature, thus 



(BO- -Tr^e>0)-Ma 



a a 2 \80 2 



Since the transverse displacement v by itself contributes 

 nothing, the increase of curvature is 



a? 



The resultant stress across any section may be resolved into 

 a radial shearing force P, a tangential tension Q, and a bending 

 moment M. On the principles of 43, 45 we have 



