962 On Circular Plates of Variable Thickness. 



are determined by (12). One of these is an additive con- 

 stant in w , since w does not occur explicitly. 



For the case of a symmetrical plate we have m' = 0, and 

 the variables separate. The equations assume the simpler 

 form : — 



+ 2 (* + ^+^'+2^)}+0. 



If, furthermore, the plate is of constant thickness, the 

 equations in w ' and v become the same : — 



r 2 v" + rv' — v = 0, r 2 w '" -f rw '' — w> ' = 0. 



This differential equation for w coincides with that obtained 

 in the usual theory (Love, p. 494). 



When w and v have been determined in the general case, 

 the displacements are given by the formulas,: — 



U = — (z — m) ipo'i ■+- hvt + ...... , 



T , J J 

 It seems possible to proceed with tnis example, as in the 

 earlier case, by considering the body forces and surface 

 tractions, and determining terms in U and w of higher order. 

 The problem of the thin circular plate under special con- 

 ditions has thus been formally treated by means of the 

 following method, apparently applicable to thin plates, shells, 

 and beams : — (1) introduction of a small parameter t to 

 represent distance from a fundamental mean surface or line ; 

 (2) expansion of the Lagrangian function T — U, the dis- 

 placements, the body forces, and the surface tractions as power 

 series in t ; (3) determination of the early coefficients so as 

 to make this integral a minimum of as high order as possible 

 in t ; and (4) determination of later coefficients so that the 

 body forces and surface tractions have the required values. 



