108 Lord Rayleigh. On the [Dec. 13, 



stiffness, and if the other coordinates be suitably chosen, the potential 

 energy of the system may be expressed 



This follows from the general theorem that V and T may 

 always be reduced to sums of squares simply, if we suppose that 

 T = tarff. 



The equations of equilibrium under the action of external forces 

 * ? *, . . are thus 



hence if the forces are regarded as given, the effect of increasing C T 

 without limit is not merely to annul ^ 1? but also the term in V which 

 depends upon it. 



An example might be taken from the case of a rod clamped at one 

 end A, and deflected by a lateral force, whose stiffness from the end 

 A up to a neighbouring place B, is conceived to increase indefinitely. 

 In the limit we may regard the rod as clamped at B, and neglect the 

 energy of the part AB, in spite of, or rather in consequence of, its 

 infinite stiffness. 



If it be admitted that the deformations to be considered are pure 

 bendings, the next step is the calculation of the potential energy 

 corresponding thereto. In my former paper, the only case for which 

 this part of the problem was attempted was that of the sphere. 

 After bending, "the principal curvatures differ from the original 

 curvature of the sphere in opposite directions, and to an equal amount,* 

 and the potential energy of bending corresponding to any element of 

 the surface is proportional to the square of this excess or defect of 

 curvature, without regard to the direction of the principal planes." 

 Though he agrees with my conclusions, Mr. Love appears to regard 

 the argument as insufficient. But clearly in the case of a given 

 spherical shell, there are 110 other elements upon which the energy of 

 bending could depend. "Thus the energy corresponding to the 

 element of surface a 2 sin0 dO d<p may be denoted by 



(2), 



where H depends upon the material and upon the thickness." 



By the nature of the case H is proportional to the elastic constants 

 and to the cube of the thickness, from which it follows by the method 

 of dimensions that it is independent of a, the radius of the sphere. 



* This is in virtue of G-auss's theorem that the product of the principal curva- 

 tures is unaffected by bending. 



