Flexure of a Flat Elastic Spring. 185 



where X, fi are the elastic constants of Lame, equivalent to 

 m—n and n, respectively, in the notation of Thomson and 

 Tait. If, further, the plate experience curvatures l/p 2 and 

 l/p 2 parallel to its edges, the corresponding flexural couples 

 are 



\ + 2fl \/°i P2/' 3 



H= 4(\ + //,W 1 | a 



\+2fJL \p 2 p 



) 



2tf 

 3 



>■ 



(5) 



Now let w denote the deviation at any point of the middle 

 surface from the cylindrical surface (of radius p) drawn 

 through the medial line, in the strained condition ; and let 

 (t be the extension of this medial line. We have evidently, 

 in the present case, 



0i = Oq + WPj 



/0!=p +W, 



1/p^-dhv/dx*. 

 The equations (1) and (4) show that 



02= — 00i) 

 and substituting from (4), (5), and (6), in (2), we get 



(6) 



A 2 d* /dhv 

 3 daAda? 



\ 1 — 2 / w\ ^ 



— + (o- +-) = 0. 



p + wj p \ pj 



(7) 



An error of the order ic/p in the values of the flexural couples 

 is clearly unimportant*, so that we may write for the last 

 equation 



^ + (W )( „, + ^) = 0. 



The boundary conditions (3) give 



d 2 w <j 



dx 1 p + w 



d (d*W a 



dx\dx 2 p + iv 



or, by a similar approximation, 



)-o, 



* This approximation may also "be justified a posteriori. Itwil appear 

 that the terms neglected are of the order h/p compared with those 

 retained. 



