218 Mr. Horace Lamb on the 



2b is large compared with the thickness 2/z, let us in the 

 first instance suppose that by a proper application of force 

 to the two ends (to be afterwards determined) it is bent so 

 that the strained form is one of revolution. As regards 

 the amount of the bending, we shall suppose only that p^ 

 the radius of curvature of the medial line or axis, is large 

 compared with h. Let the band be divided into rectangular 

 elements by two systems of normal sections respectively 

 perpendicular and parallel to its length. The stresses across 

 any section of the former system will reduce to a tension P,. 

 and a couple G, both reckoned per unit length ; whilst 

 across any section of the second system we shall have 

 a tension Q, a couple H, and a shearing force Z in the 

 direction of the normal. All these quantities are functions 

 only of the distance (x, say) from the medial line. Con- 

 sidering the equilibrium of a rectangular element of 

 breadth dx, and resolving parallel to x, we find 



dx ' 

 and since Q vanishes at the free edges, it follows that 



Q = (1) 



everywhere. Again, resolving along the normal, we have 



ax p 

 and taking moments about a parallel to the medial line, 

 dK 



Hence 



dx " 



dm p n , • 



-d^-~r (2) 



The remaining boundary conditions are evidently H = 

 Z = 0, that is 



H-0, § = 0. (3) 



