106 Lord Rayleigh. On the [Dec. 13, 



tude, the solution contains two arbitrary constants ; but if a pole be 

 included, as when the shell is in the form of a hemisphere, one of the 

 constants vanishes, and the type of deformation is wholly determined, 

 without regard to any other mechanical condition, to be satisfied at 

 the edge or elsewhere. It will be convenient to restate, analytically, 

 the type of deformation arrived at [equation (5)]. If the point 

 upon the middle surface, whose coordinates were originally a, 0, 0, 

 moves to a -f- r, + d0, + 0, the solution is 



= A tan 5 J0 cos s0 



16 = - A sin 6 tan'^0 sin 50 V .......... (1), 



&r = Aa (s + cos 0) tan 5 J0 sin s0 



6 being the colatitude measured from the pole through which the 

 shell is complete. Any integral value higher than unity is admis- 

 sible for s. The value and 1 correspond to displacements not 

 involving strain. 



In a recent paper* Mr. Love dissents from the general principle 

 involved in the theory above briefly sketched, and rejects the special 

 solutions founded upon it as inapplicable to the vibration of thin 

 shells. The argument upon which I proceeded in my former paper, 

 and which still seems to me valid, may be put thus : It is a general 

 mechanical principlet that, if given displacements (not sufficient by 

 themselves to determine the configuration) be produced in a system 

 originally in equilibrium by forces of corresponding types, the result- 

 ing deformation is determined by the condition that the potential 

 energy of deformation shall be as small as possible. Apply this to an 

 elastic shell, the given displacements being such as not of themselves 

 to involve a stretching of the middle surface.^ The resulting 

 deformation will, in general, include both stretching and bending, 

 and any expression for the energy will contain corresponding terms 

 proportional to the first and third powers respectively of the thick- 

 ness. This energy is to be as small as possible. Hence, when the 

 thickness is diminished without limit, the actual displacement will be 

 one of pure bending, if such there be, consistent with the given con- 

 ditions. Otherwise the energy would be of the first order (in 

 thickness) instead of, as it might be, of the third order, in violation 

 of the principle. 



It will be seen that this argument takes no account of special 

 conditions to be satisfied at the edge of the shell. This is the point 

 at which Mr. Love concentrates his objections. He considers that 



* " On the small free Vibrations and Deformation of a thin elastic Shell," ' Phil. 

 Trans.,' A, 1888. 



f Phil. Mag.,' March, 1875 ; ' Theory of Sound,' 74. 



J There are cases where no displacement (involving strain at all) is possible 

 without stretching of the middle surface, e.g., that of the complete sphere. 



