[ 953 ] 



CI. Circular Plates of Variable Ihichiess. By George 

 D. Birkhoff, Ph.D., Professor of Mathematics, Harvard 

 University *. 



AS far as I am aware, only plates and shells of constant 

 thickness have been considered in the theory of elas- 

 ticity. The aim of the present note is to develop a method 

 applicable when the thickness is variable |. The method is 

 here applied to thin circular plates, nearly plane, and clamped 

 at the outer edge. 



Case I. — The Incomplete Symmetrical Plate under 

 Radial Pressure. 



Suppose that ordinary cylindrical coordinates (fig. 1) 

 are adopted, with the axis of z along the axis of the plate. 



Fig. 1. 



The plate is of inner radius a and outer radius b, and is 

 symmetrical about the plane ^ = 0. Let a be proportional 

 to the thickness of the plate, so that z = ta and,s=— £a may 

 be taken as the equations of the upper and lower bases 

 respectively. Here t is a small parameter, since the plate is 

 thin. 



Radial forces are applied to the inner edge, so that points 



* Communicated by the Author. 



t I am greatly indebted to Mr. Carl A. Garabedian, of Harvard 

 University, for able assistance in carrying through some of the calcu- 

 lations and for verifying those I have made. Mr. Garabedian is under- 

 taking the consideration of more general problems by this method. 



