956 Prof. G. D. Birkhoff on Circular 



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

 formed. The principal part of W takes the form W^, where 



n 



This integral is to be made a minimum subject to the boun- 

 dary conditions 



U (a) = e, U (&) = 0. 

 Accordingly our problem is reduced to a simple problem in 

 the Calculus of Variations. The condition 8Wi = gives 

 at once 



dr'dVo' au 



= 0, . . . , . (2' 



where <l> is the integrand in (1). Writing out this equation 

 in full, we obtain 



^^(X + ^Uo' + X— °]ar-r2(\ + /A )^+XU ']« = 0. 



Consequently U must satisfy the folllowing differential 

 equation : — 



W*¥-%~-U" J+ J!&)% ■ (3) 



where 2h=2tu stands for the thickness of the plate. 



In the case of a plate of constant thickness, li is constant 

 and 7i' = 0, and (3) reduces to a well-known form (Love, 

 p. 141). 



Thus far we have determined the displacements to be 



U=U„0)+TJ 2 < 2 + ...., (4) 



*=-5^( U » , + T> + V+v/- • • ( 5 



where it is to be remembered that the accent on z has been 

 suppressed. 



We proceed to determine U 2 from the fact that the body 

 forces F r , ¥ z must vanish. We have (Love, p. 141) : 



(6) 



X+'2fx~d 2 w X + fj, ( JV . U\ 



+ / /i|!?, + ^) = o. . (7) 



\r or or~ / 



