960 Prof. a. D. Birkhoff on Circular 



that the energy will be of order t 3 . Thus we must take 



U = 0, w; 1 = i« 1 (r), U, 



zwj+p(r), 



where p(r) is arbitrary. None of these requirements violate 

 the boundary conditions. 



The physical significance of the conditions thus far obtained 

 is immediate ; the displacement is transverse, and such that 

 filaments perpendicular to z=0 remain perpendicular in the 

 displaced position and are not compressed transversely. 



When these conditions are imposed, it is easy to see that 

 the energy is of the order tK More precisely, the leading 

 term in W now becomes 



7Tt 



ZW + p 





3 P p [(Xf2^)j"-. 2 W'+p' + 



±(-zw o ''+p')^0j\rdz 



w ' + p\'d< 



[{~ ZW T~) 



r J ~dz ' ' c ' d~ _ 



The part of the integrand in braces may be written as 



dr. 



zw " -fp'-f - 



■zw '+p j_ B^ 2 T 



~dzJ 



+ 



■(-W+pr+Jc 



Since U 2 appears only in the last term, and since W is to 

 be a minimum, we choose 



We proceed next to the choice of ~, and find (by the 

 same method as employed in the first example) : — 



"bw 2 

 ?>z 



X+2p, 



( — 



(-zw "+p' 



—zwo' + p 



IV 2 - 



X 



^ w Q " — zp' + 



iv —zp 



+ s{r) 



s(r) arbitrary. It may be noted that the choice of U 2 , iv 2 

 do3S not interfere with the boundary conditions. 



