( '542 ) 



§ 3. T shall consider in the first place an infinitely thin spheri- 

 cal shell of radius a, charged, in the state of e(]uilibrium, to a 

 uniform surface-density u. The surface-density of the ponderable 

 matter itself will be denoted by g. We shall suppose that the points 

 of the shell can only be displaced along its surface, that the elements 

 carry their charge along with them, and finally that, after a displa- 

 cement, each element is acted on by an elastic force, which is 

 brought into play merely by the displacement of the element itself, 

 and not by the relative displacement of adjacent elements. 



When the motions are infinitely small, the elastic force may ho 

 taken proportional to the displacement a. Lot it be 



— /c~ a 



per unit area, the constant /- having the same value all over the sphere. 

 The only connexion between the different parts of the shell will 

 consist in their mutual electric forces. If the wave-length of the 

 emitted radiations be very large in comparison with tlie radius of 

 the sphere, we have merely to consider the ordinary elect rostatic 

 actions, depending solely on the configuration of the system. Ilence 

 there will be no resistances proportional to the velocities, and con- 

 sequently no damping. In fact, it is well known that the damping 

 which, in some degree, must always be caused by the loss of 

 energy, accompanying the radiation, may be neglected when the 

 wave-length is vciy much larger than the dimensions of the vibra- 

 ting system. 



§ 4. In the ahsence of magnetic forces the shell can vibrate in 

 the following way. 



Let i/, be a suiface haimonic of oider //. Then the displaecniciit 

 of a point of the sphere is 



^ T/ • ^'^ 



Here / is the direction in the surface in which )'/, increases most 

 raiiidh, and is to be reoardcd as a vector in this direction I. 



The coefficient p is the same all over the sphere; it has the form 



q cos {vkt + (■) , (2) 



so that ?//, is the frequency of the vibrations. 



