1888.J Bending and Vibration of thin Elastic Shells. Ill 



three dimensions, as given in Thomson and Tait's ' Natural Philosophy,' 

 694, are 



na = S, nb = T, no =U ........ (9) 



l 

 V .............. (10), 



= R-<r (P-fQ)J 



where 



The energy w, corresponding to the unit of volume, is given by 



+ 2 ( m - n )(fg + ge + ef)+n(a? + V + &) ...... (12). 



In the application to a lamina, supposed parallel to xy, we are to 

 take R = 0, S = 0, T = ; so that 



g <r T -, a = 0, b = 0. 



1 a 



Thus in terms of the elongations e, /, parallel to #, i/, and of the 

 shear c, we get 



We have now to express the elongations of the various laminae of a 

 shell when bent, and we will begin with the case where r = 0, that is, 

 when the principal planes of curvature remain unchanged. It is 

 evident that in this case the shear c vanishes, and we have to deal 

 only with the elongations e and f parallel to the axes. In the section 

 by the plane of zx, let s, s' denote corresponding infinitely small 

 arcs of the middle surface and of a lamina distant h from it. If ^ 

 be the angle between the terminal normals, s = p^, s' = (/> x + Ji) i^, 

 s' s = h Y^. In the bending, which leaves s unchanged, 



Hence 



e = 



and in like manner/ = h8 (l//> 2 ). Thus for the energy U per unit of 

 area we have 



* M is Young's modulus, a is Poisson's ratio, n is the constant of rigidity, 

 and (m \n) that of cubic compressibility. In terms of Lame's constants (A, fi), 



