of an Elastic Plate under Normal Pressure. 117 

 Solving these for H and K, we get 



a sinh 6 cos 6 — cosh sin 6 ,„ 9 s 



. • (73) 









m 2 a sinh 26 + sin 26 









K= 



a cosh 6 sin 6 4- sinh # cos 



<9 





" ra 2 a sinh 26 -+ sin 2# 





wherein 



it 



should be noticed that 











cr crA 





Thus 



^^~\/3(l-o- 2 )' 





lOi- 





a 



1 



T^ r— 7 TTi X 





(sinh (9 cos # — cosh 6 sin 0)cosh ??!# cos my) . 

 -f (cosh sin 6 + sinh (9 cos 0)sinh *riy sin wi#J' 



and w= k Wi (75) 



la 



As a particular case suppose ml, and therefore also my, 

 is a small fraction. Then an approximate value of w± 9 

 obtained by expanding the hyperbolic and circular functions, 

 is 



-=-:{i* 2 -M (76) 



This shows that the section of the middle surface by the 



yz plane is nearly a circle of radius - . Thus our result 



gives the well-known anticlastic curvature of a bent rect- 

 angular rod. 



Let us next suppose that ml is large, so that we may 

 assume 



cosh 0= sinh = ie e ; sinh 20- J e 26 . 



Then 



nr 



_ ae~ e J (cos 6 — sin 6) cosh my cos my) (il\ 



l2<x ( + ( cos ^ + sin 6) sinh my sin my) 



In order to find the form of the surface near the free 

 edges we can put 



y=i-y' 



and assume that y' , the distance from the nearest free edge, 



