1888.] Bending and Vibration of thin Elastic Shells. 109 



I did not, at the time, attempt the further determination of H, not 

 needing it for my immediate purpose. Mr. Love has shown that 



H = **fcs ...................... (3), 



where 2Ji represents the thickness, and n is the constant of rigidity. 

 Why n alone should occur, to the exclusion of the constant of com- 

 pressibility, will presently appear more clearly. 



The application of (2) to the displacements expressed in (1) gave 

 [equation (18)] 



V = 2:7-2 (s 3 s) A, 2 *H sin- 3 tan 2 'J0<Z0 ........ (4), 



9 being the colatitude of the (circular) edge. In the case of the 

 hemisphere of uniform thickness 



Y = iTrHS (s*-s) (2s 2 -l) A/ .............. (5). 



The calculation of the pitch of free vibration then presented no 

 difficulty. If a denote the superficial density, and cospt represent the 

 type of vibration, p% corresponding to s = 2, p 3 to s = 3, and so on, it 

 appeared that 



/TT /TT 



x .5-2400, ^ 3 = ^--F x 14726, p 4 = V- x 28-462 ; 



so that 



= 2-8102, P Jp 5 = 5-4316, 



determining the intervals between the graver notes. 



If the form of the shell be other than spherical, the middle surface 

 is no longer symmetrical with respect to the normal at any point, and 

 the expression of the potential energy is more complicated. The 

 question is now not merely one of the curvature of the deformed 

 surface; account must also be taken of the correspondence of normal 

 sections before and after deformation.* A complete investigation 

 has been given by Love ; but the treatment of the question now to be 

 explained, even if less rigorous, may help to throw light upon this 

 somewhat difficult subject. 



In the actual deformation of a material sheet of finite extent there 

 will usually be at any point not merely a displacement of the point 

 itself, but a rotation of the neighbouring parts of the sheet, such as a 



* An extreme case may serve as an illustration. Suppose that the bending is 

 such that the principal planes retain their positions relatively to the material sur- 

 face, but that the principal curvatures are exchanged. The nature of the curvature 

 at the point in question is the same after deformation as before, and by a rotation 

 through 90 round the normal the surfaces may be made to fit ; nevertheless the 

 energy of bending is finite. 



