68 Mr Ibbetson, On the small free normal vibrations [Jan. 28, 



those equations was not the object I had in view, so I did not 

 consider that I was required to pause in order to state it explicitly 

 in the inverse notation. 



On referring to the Note, it will be seen that I was discussing 

 the oscillations of a rocking body whose centre of gravity is a very 

 short distance on the unstable side of the critical position, and for 

 which therefore c is very small: and the result is obtained by con- 

 sidering the small change produced in the critical period and the 

 critical motion by the small amount of apparent instability. 



But when c becomes at all large the character of the motion is 

 quite altered : the flanking stable positions are now some distance 

 from the centre of the oscillation, and we can no longer neglect 

 higher powers of 0. 



Thus Mr Greenhill's statement of the solution is not applicable 

 to the problem when c is not small: and when c is small, it is 

 unnecessarily complex. The reduction to a simple approximate 

 form corresponding to that which I gave in the Note (in which the 

 only modulus that occurs is sin 45°) would involve differentiation 

 of the inverse functions with respect to the modulus : and I cannot 

 see that such a process would lead to results simpler or more 

 calculable than those which I obtained. 



(3) On the small free normal vibrations of a thin homogeneous 

 and isotropic elastic shell, bounded by two confocal spheroids. By 

 W. J. Ibbetson, B.A. 



[I-] 



So far as I am aware no attempt has yet been made to solve 

 a problem of this kind by the direct use of curvilinear coordinates, 

 other than polars. 



The present paper is limited to the case in which the shell 

 vibrates in such a manner that its surfaces always remain spheroids 

 confocal with their unstrained forms. This case is interesting 

 because it is the only possible form of motion unaccompanied 

 by shear in the substance of the shell, and because the results 

 admit of numerical calculation. 



It is obvious that throughout the motion the two systems of 

 orthogonal surfaces (planes through the axis, and hyperboloids of 

 one or two sheets, according as the shell is oblate or prolate) 

 always remain the same. 



The corresponding problem in polars is therefore to investigate 

 the radial vibrations of a spherical shell of uniform thickness re- 

 presented by an harmonic of zero order. 



We shall see that the solution for the sphere can be readily 

 deduced from the results obtained for the spheroid. 



