958 



Prof. G. D. Birkhoff on Circular 



Hence we find along the x axis 



eos o, v) = — 77-^-70 ' cos (y> ^ = °> cos ( z > v ) = 



v/l-h^V 



Ksr. 2. 



v/l + *V 2 ' 



Z=tC( 



Hence the conditions that the tractions vanish on the free 



surfaces become 



-to'X^ + X^O, (9) 



-t*'Zx + Z g = (10) 



As we proceed to higher order terms, using (6) and (7), 

 the relations (9) and (10) may be expected to play an im- 

 portant role in furnishing differential equations to determine 

 the arbitrary functions that enter. In the present case there 

 is no constant term in either left-hand member; and by 

 virtue of (3) the terms in t also vanish. 



Thus the surplus body forces per unit of volume vanish 

 up to order t, and the surplus surface tractions per unit of 

 area vanish up to order t 2 . Hence the total surplus applied 

 force is of the order t 2 , whereas the total given radial force is 

 of order t. It seems clear that when the surplus applied 

 forces are removed, no sensible change occurs in the radial 

 or axial displacement, and that we have a solution of our 

 problem. 



It remains to compute the principal part of the radial 

 pressure on the inner edge of the plate. Here we make use 

 of Saint Venant's principle and obtain the resultant traction, 

 Xx(r, z), as 



« x 2n £ta{a) 

 ^0 J-ta(a) 



X x (a, 0) adr cW = 



|U '(a) 



X U (a) j 

 2(X-\-/jl) a j 



If we desire to take £=1 throughout, so that the equations 

 of the bases are : = «and z= — a, then t disappears as an 

 explicit parameter, but the terms of the solution are still 

 ordered according to the powers of the natural parameter, 

 namely the ratio of the thickness of the plate to its diameter. 



