r dz dr, 



Plates of Variable Thickness. 955 



order t. Hence if W is to be a minimum, it is clear that we 

 must make the leading term of W, namely 



W d ('• r3U - 

 tj a UL3~-J 



vanish if possible. But this quantity vanishes if, and only 

 if, U = U (r), and by thus restricting U the stated boundary 

 conditions are not violated. 



When U is thus restricted, the double integral W will 

 involve no negative powers of t, and will have as leading 

 term : — 



U ~dwp 2 

 + '3* 



r dz dr } 



5 r*rr|(\+2 /t )[u„'+5 



where accents denote differentiation with respect to f. The 

 part of the integrand in braces may be written as the sum 

 of squares : — 



x[u/ + ^ + ^]%2,[u * + ^ + (|^J. 



We turn next to the choice of -— 1 , which is an arbitrary 



function of r and z still at our disposal. If w T e call this 

 variable x, the above expression involves x in two terms of 

 the form 



Elementary calculation shows that for m=- ^- , the f ore- 



J X + 2/a' 



going expression has the minimum value ^ m 2 . Hence 



we must take a, + w/x, 



"div x —\ / TT , U 



~dz \-\-2fi 



(w+ V> 



or 



^=^{ U °' + r) + Sir) ' 



s{r) arbitrary. But w(r, 0)=0 by (B) ; hence 5(7*) =0. 

 This choice of w 1 does not interfere with the boundary con- 

 ditions. 



The terms written above now reduce to give 



» 



€t^{^^--^y^y^ 



