Plates of Variable Ihickness. 961 



The integral written above thus becomes 



+M-~~<'+ 2 >')( =£ v ±£ )] J-'*"*. 



and an integration with regard to z can be explicitly per- 

 formed. To this end it is convenient to introduce a new 

 variable z\ where 



z = 7n + z\ /3 = m4-h, a=m—h ; 



at the same time we replace p by v 9 where 



so that hvt is the radial displacement for the mean surface 

 z=m. With this notation the principal part W 3 t 3 of W has 

 a coefficient. 



It is to be observed that — , -7- are independent of the 



11 li 



thickness of the plate. 



The integral is to be made a minimum subject to the 



boundary conditions 



w (Q) = e, w (a) = 0, w o '(a) = 



v(0)=v(a)==0.' 



The form of W shows that Wo(Q) vanishes also; otherwise the 

 integral would be infinite. 



Our problem is now one in the Calculus of Variations, and 

 the condition &W 3 = gives us 



d '&&__'<& =0 ^B^_3^ =0 (13) 



dr"dw " 'divo ' dr'dv 1 ~dv ' ' 



where <E> is the integrand in (11). We will not write out 

 these equations. The system of two equations is of the fifth 

 order. The five arbitrary constants in the general solution 

 Phil. Mag. S. 6. Vol. 43. No. 257. May 1922. 3 Q 



