118 Dr. J. Prescott on the Equations of Equilibrium 

 is small compared with I. On doing this we get 



ICi 



__ ere' 6 ( i(cos0 — sm 0)e e - m *' cos(0 — my')) 

 " m 2 a \ + ±(cos0 + sm6)e e - m y'sm(0 — my')) 



e~ m v'(cos my' — sin my') 



G 



2m?t 



Thus we see that, when a broad rectangular plate is bent 

 into a surface of revolution, small corrugations are formed 

 on the bent surface, each crest and each trough forming a 

 circular arc about the axis of revolution. However wide 

 the plate may be, the maximum amplitude of the corrugations 



is approximately \J «o-/*, and the amplitude is smaller the 



nearer the wave lies to the middle circular arc of the bent 

 plate. 



When the plate is narrow, so that it approaches what we 

 may call a beam or rod, only one small length of the 

 oscillatory curve is comprised on the breadth of the plate, 



and this may be regarded as a circle of radius - . 



The extreme cases to which we have referred occur when 

 ml is large and when ml is small respectively ; that is, 

 when I 2 is large and when I 2 is small compared with ah. 

 When I 2 and ah are of the same order the complete 

 expression for w\ must be used. 



If we had started with a sheet whose unstrained middle 

 surface was that of a piece of a cylinder bounded by two 

 generators and two circular arcs, it is easy to see that the 

 preceding investigation can be adapted to give the strain 

 when this cylinder is bent still further by forces and couples 

 applied along the edges of the generators. If the radius of 

 the cylinder before strain was b, and 'the approximate radius 



after strain a, then we have only to replace - in the foregoing 



work by I T ). If I 2 is small in comparison with -, 



J \a b) b — a 



the longitudinal section of the bent middle surface is 

 approximately a circle with curvature a I -); whereas 



if I 2 is large in comparison with y , there are many 



