220 



Mr. Horace Lamb on the 



An error of the order wjo in the values of the flexural 

 couples is clearly unimportant,* so that we may write for 

 the last equation 



^^(i-" 2 )(^op) = 0. (7) 



The boundary conditions (5) and (6) give 



dhv 



dx 2 p + 7£> 



= 0, 



or, by a similiar approximation, 

 d 2 w _(t _ 

 dx 2 p 



d*w 



d (d 2 w _ o- \ _ a 

 dx\ dx 2 p + w) 



0,) 



dx 3 

 Let us write for shortness 



= 0, 



," = ;? 



3(i-* 2 ) 



(8) 



(9) 



4 hy 



The proper solution of (7) is then 



w + (T p = Acoswaxoshw^ + Bsin;;/^sinh;;/^ ■ (10) 

 and the conditions (3) to be satisfied at the edges x=+b 

 give 



leading to 



- Asin//z£sinh//^ + Bcoswfcosh//^ = — =- 



my 



A(cosmbsmhmb + s'mmbcoshmb) 



+ B(sinmbcoshmb 



- cosmbs'mhmb) = 0/ 



A = 



B 



a smmbcoshmb — cosmbs'mhmb 

 m 2 p ' sinh2;«^ + s'm2mb 



a sinmbcoshmb + cosmbsinhmb 

 m 2 p ' 



(«) 



(12) 



sinh2;«^ + sin2w^ 

 The condition that w = Q when x=0 gives 



<T p = A, (13) 



and the value of w is thus completely determined. 



*.This approximation may also be justified a posteriori. It^ will appear that 

 the terms neglected are of the order hip compared with those retained. 



