108 Dr. J. Prescott on the Equations of Equilibrium 

 Also the principal curvatures of the middle surface are 



Consequently 



d 2 w , 1 dw 

 -j-z and — =—. 



ar r dr 



*>-° {%+',%} 



M, = C 



HI- 



1 dw d 2 io~\ 

 dr ^^ ' 



dr 2 



(23) 

 (24) 



Again 



Q = 0. 



_ 9 d 2 w 1 dw 

 dr* r dr 



It follows that 



=M(4:h (25) 



F *=-S1(VS 



C d % 1 d / ,^\ \ > 

 ~~21id~r\ rlr\1r) j | 



(26) 



J 



F 2 = 



Finally, the differential equations (12) and (14) become 



1 dw d 2 w 



d 

 dr\ dr 



dw 

 ~dr 



d 2 w 

 dr 2 



(27) 



r r dr dr z 



' r dr\ drj * 

 r—,A ^7-t-.] ( d 2 io d(b dw d 2 d>\ 



p^-TSl^^ • • (28) 



Problem 1. — To find the stresses in a circular plate which 

 Is bent into the form of a piece of a spherical surface, of 

 small solid angle, bounded by a circle. 



that is. 



where 

 and 



that is. 



2 EA 3 



3 1 



