AND DEFORMATION OK A THIN KLASTIC SMKLL. ' ' ' 



.T:iini:iry 18th, morn., 1*7S, c|iir-ti<>n 77. To test this assumption involved the investi- 

 gation of Artt. 7. .-. an. I die n-Milt is tli.it it is only in the case of a sphere supposed 

 imstretched that the potential i-nrr^y lias this form. This is the case treated by 

 Lord liAYi.KKiii, hut his method still fails, for a complete sphere cannot be bent 

 without stretching, while, if the sphere be incomplete, the conditions which hold at a 

 free edge cannot be satisfied ; this is explicitly proved in Art. 14. 



A general result is derived from the consideration of the functions expressing the 

 kinetic and potential energies, Q 2 only being retained. Both these functions are 

 proportional to the thickness of the shell, and thus the periods of vibration are inde- 

 pendent of the thickness. That this result holds for a complete thin spherical shell 

 vibrating in any manner has been demonstrated by LAMB (' London Math. Soc. Proc.,' 

 vol. 14, 1882, p. 52). His equations (7) and (9) when reduced are independent of the 

 thickness. 



Two general results are obtained without solution from the equations of motion. 

 The first is, that vibrations involving displacement along the normal only are impos- 

 sible except in the cases of the plane, complete sphere, and infinitely long circular 

 cylinder. IBBETSON'S treatment of the problem appears to assume (I) inextensibility, 

 (2) the incorrect formula for the energy, (3) normal displacements. The other result 

 is that any surface of revolution can execute purely tangential vibrations which are 

 symmetrical with respect to the axis of revolution, and in which the motion is purely 

 torsional, or perpendicular to the planes through the axis. These must not be 

 confounded with the familiar vibrations of finger-bowls, which are most probably a 

 type with two nodal meridians.* 



The theory of the vibrations of a thin spherical shell bounded by a small circle is 

 an interesting example of the general theory of vibrations of an elastic solid. In an 

 infinite solid there are two types of vibratory motion, the longitudinal and the 

 distortional, both of which are propagated as waves. In a bounded solid this state of 

 things is modified by reflexions at the bounding-surfaces, so that the purely longitu- 

 dinal and purely tangential waves do not in general exist separately. Again, in all 

 cases of displacement in one direction only, as in the vibrations of strings, bare, and 

 plates, there may be displacements in different directions which are independent of 

 each other, with their corresponding nodal lines or points. This also is modified in 

 the general solid. The types of vibration, for example, of a portion of a spherical 

 shell bounded by a small circle are partially made out in this essay. One immediate 

 result is that there are in general no nodal lines, properly so called. Tn any type the 

 displacement along the parallels vanishes at one set of meridians ; the other displace- 

 ments vanish together at another set of meridians. These sets are ranged at equal 

 intervals round the sphere. There appears to be good reason to suppose generally 

 that the corresponding proposition will not obtain with reference to nodal parallels. 

 The establishment of the fact would require a solution of the general frequency cqua- 



RATI.EIOH, Sound,' vol. 1, Art, 234. 



MDCCCLXXXVIII. A. 4 A 



