954 Prof. Gr. D. Birkhoff on Circular 



for which r = a undergo a displacement of amount e. The 

 plate is clamped at the outer edge. Thus, if U and w denote 

 the radial and axial displacements, the boundary conditions 

 are 



(U(a,0) = e, U(M) = 0, 



\ w[o,0)= — ^ - — 0. 



I Or 



Obviously such a plate is unstable, with a tendency to 

 buckle. 



It is a cardinal fact of the theory of elasticity that the 

 actual displacement of the plate will be such as to yield the 

 minimum potential energy consistent with the constraints 

 imposed (Love, ' Theory of Elasticity,' third edition, p. 169). 



By Love (pp. 99-100, 141) this potential energy W is 

 given as follows : — 



LAd£ or J r or r Oz or Oz J j 



It is natural to introduce a new variable z\ such that z = tz'. 

 When we replace z by z' and afterward suppress the accents 

 there results : — 



(A) 



— m«M¥,^+m' 



rdzd'i 



+ / TV oz * or J r 3r * r *z t 3r Sc J J 



Our assumption will be that all of the quantities involved 

 can be expanded in ascending powers of t — in particular that 



U=U + U 1 *+U 2 * 2 + , 



W — W + Wit + w 2 t 2 + 



It is to be noted that if in (A) we replace z by — z and 

 w by —to, or t by —t and w by — w, the double integral is 

 altered at most in sign. Since these transformations do 

 not disturb our boundary conditions, and since W has a 

 unique minimum, the special relations 



(B) U TO (r, z) — U ra (r , — z) = w m (r, z) ■+■ w m (r, — z) = 0, 



m = 0,l,2,..., 



(B') U 2m+ i(r, z)=w 2m (r, z) = 0, m =0,1,2, , 



must obtain. 



The case of a plane plate shows that the energy is of 



