114 Dr. J. Prescott on the Equations of Equilibrium 



of edges being circular arcs. To find the form of this 

 surface. 



Let the xy plane be parallel to the tangent plane to 

 the bent surface at its middle point. It is less troublesome 

 to take the xy plane parallel to this tangent plane than 

 actually coincident with it. Also let the y-axis be parallel 

 to the axis of the surface of revolution, and the --axis drawn 

 towards the concave side of the cylinder. 



The sections of the middle surface perpendicular to the 

 ?/-axis are circular arcs with nearly equal radii. Let p be 

 the radius of the arc at distance y from the xz plane, and let 

 ■a denote the distance of the axis of revolution from the 

 xy plane. Thus a is an approximate value of the radius of 

 the circular arcs, since it is understood that the xy plane 

 nearly coincides with the tangent plane at the centre of the 

 bent middle surface. 



Now let p = a-\-w 1 , . . '. . . . (51) 



where, clearly, w Y is a function of y only. 



Then the displacement of any point of the bent plate 

 (fig. 6) at a small angular distance from the origin is 

 approximately 



2p 



= —10! + 



Fur. 6. 



2a 



(5! 



Hence 



Wv> 



I* 

 ~dx 



V 2 w= 

 V 4 w=- 



\ 'dhv 

 a ; V' ! 



1 d 2 Wj 



a dy 2 

 d*w x 



W 



cPw^ yw 



dtf ' ~dx~dy 



-0. 



(53) 

 (54) 

 (55) 



