of an Elastic Plate under Normal Pressure. 109 



Taking the xy plane touching the middle surface of the 

 bent plate at its middle point, the deflexion is approximately 



r 2 



W= 2c> 



c being the radius of the sphere. 



Let a be written for the radius of the plate. 

 Then equation (27) becomes 



dr \ dr J c 2 ' 



Integrating this once, we get 



dyjr r 2 TT 



The constant H must be zero because the other two terms 

 in the equation are clearly zero when r=0. Therefore 



- d^jr r 



~dr~ = ~2?' 



whence yl/ = — ^~^{r 2 — b 2 ), 



b 2 being the constant of integration. 

 Now we have got 



r dr 



{%)—&-»>- • • • <?> 



Integrating again, and omitting the new constant, which 

 is obviously zero, we get 



r *r= -£?&**- W (30) 



Now the mean tensions are 



^*%-JS*W--*>> ■ ■ • (3D 



Let us now suppose that the radial tension is zero at the 

 rim of the plate. That is, 



Pi = where r = a, 



or 2b 2 -a 2 = 0, 



which gives the constant b 2 . 



