110 Lord Rayleigh. On the [Dec. 13, 



rigid body may undergo. All this contributes nothing to the energy. 

 In order to take the question in its simplest form, let us refer the 

 original surface to the normal and principal tangents at the point in 

 question as axes of coordinates, and let us suppose that after deforma- 

 tion, the lines in the sheet originally coincident with the principal 

 tangents are brought back (if necessary) to occupy the same positions 

 as at first. The possibility of this will be apparent when it is re- 

 membered that in virtue of the in extensibility of the sheet, the angles 

 of intersection of all lines traced upon it remain unaltered. The 

 equation of the original surface in the neighbourhood of the point 

 being 



that of the deformed surface may be written 



.(7). 



In strictness (p l + 3/i)"~V(/>2 + ^Pz)~ l are * ne curvatures of thesections 

 made by the planes x = 0, y = ; but since principal curvatures are 

 a maximum or a minimum, they represent with sufficient accuracy 

 the new principal curvatures, although these are to be found in 

 slightly different planes. The condition of inextensibility shows that 

 points which have the same x and y in (6) and (7) are corresponding 

 points, and by Gauss's theorem it is further necessary that 



. 



Pl P2 



It thus appears that the energy of bending will depend upon two 

 quantities, one giving the alterations of principal curvature, and the 

 other T depending upon the shift (in the material) of the principal 

 planes. 



In calculating the energy we may regard it as due to the stretchings 

 and contractions under tangential forces of the various infinitely thin 

 laminae into which the shell may be divided. The middle lamina 

 being unstretched, makes no contribution. Of the other laminae, the 

 stretching is in proportion to the distance from the middle surface, 

 and the energy of stretching is therefore as the square of this 

 distance. When the integration over the whole thickness of the shell 

 is carried out, the result is accordingly proportional to the cube of 

 the thickness. 



The next step is to estimate more precisely the energy correspond- 

 ing to a small element of area of a lamina. The general equations in 



