AND DEFORMATION OP A THIN ELASTIC SHELL. 



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and w is given by the equation 



Substituting from o- lf <r. 2 , CT = 0, we find 



a 



ir 



Now, 

 Thus, 



a\ 



i '/< 



__ 



sin 8 5 



i ar. a /. a a\/ \i 



= - a ^ si" ^ m ^ 5^ "/i 

 sin 4 80 L \ Sff/ \sin ^/J 



= ^ 8m P 



sin 



-a 



~ 81U P COS 



^" i 

 -3 + M 



* 



Hence, * 2 = Xj.* 



The boundary-conditions arising from the terms in w, &v in Art. 11 are now 



2X(1 -O-)KJ 2/t(l -<r)c 1 = 0, 

 cr)^ 2X(1 <T)K! = 0; 



since X 2 -f- p? = 1, and <r is positive, the only way of satisfying these equations 

 is to take ^ = 0, * 2 = at the edge. 

 Now, 



^* ^ ^ / * ~ 



.2 r , 3 j d / 1 < ' 



And, as shown by Lord RAYLEIGH, 



r fl" 



u = 2 sin ^ A, tan* - 



v= 's'sin^lA^tan'- 



2 L -Jin<* 



u? = ' 2* (s + cos 0) |A, tan' - 



This might have been written down at once by the aid of GAUSS'S deformation theorem. 



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