546 MR. LOVE ON THE VIBRATIONS, KTC., OF A THIN ELASTIC SHELL. 



tion, and this I have not been able to effect. One case, however, is readily solved, 

 and that is where the displacement is symmetrical with respect to the pole of the 

 sphere. It appears here that the vibrations divide themselves into two types, one 

 purely tangential with displacement along the parallels, the other partly radial and 

 partly consisting of displacements along the meridians. There are no nodal meridians. 

 In the purely tangential vibrations there exists a series of nodal parallels, whose 

 number corresponds to the type of vibration. The intervals for the various tones are 

 each of them nearly a fifth. In the partly radial vibrations the radial displacement 

 vanishes at one set of small circles, and the tangential displacement at another set. 

 The number and position of the nodal circles for the purely tangential vibration 

 coincide exactly with the number and position of the circles along which the 

 tangential displacement vanishes in the corresponding partly radial mode. The 

 vibrations of the two types belong to different normal modes of vibration, and have 

 different frequencies. If we like to extend the meaning of " nodal lines," so as to 

 include the small circles just referred to, then we may state another result in the 

 form that for partly radial vibrations there are two periods and modes of vibration 

 which have the same set of " nodal lines." The tones of one of these sets are all 

 very near together ; those of the other set are separated by intervals nearly the same 

 as for a harmonic scale. 



A discussion of the vibrations of an elastic shell in the form of a circular cylinder 

 closed at one end by a rigid disc perpendicular to its axis leads to similar conclusions 

 as to types of vibration and their definition by nodal lines. 



It is unfortunate that solutions of the frequency equation for the case of two 

 "nodal" meridians dividing the shell into four equal portions could not be obtained, 

 as these probably include the gravest mode of vibration of which the shell is capable. 

 The tones of the symmetrical vibrations discussed are very high, and the theory in its 

 present state cannot easily be tested by experiment. There is, however, one result 

 which would seem to admit of practical verification, viz., it is found that, for similar 

 thin shells, the frequency is independent of the thickness, and varies inversely as the 

 linear dimension. 



