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XIV. On the Stresses in a Heavy Spherical Shell, 

 By R. F. GwYTHER,'lf.J..* 



f I THIS represents a portion of a paper read before the 

 A Manchester Literary and Philosophical Society on the 

 7th of March. Stated shortly, the object of the paper is to 

 show that the formal solution of the statical state of stress in 

 a heavy body of an assigned shape is in certain cases 

 determinable ; that, in such a body, the determination of the 

 statical state of stress should precede any investigation of the 

 elastic stress in the body ; and that, if the statical state of 

 stress does not satisfy the conditions for being purely elastic, 

 the equations for the elastic state of the body must be 

 modified so as to conform with the statical conditions. 



It is proposed to give here only the formal solution in a 

 heavy spherical shell or bowl. The solution for a long heavy 

 circular cylindrical tube lying horizontally is worked out in 

 the paper, but a single instance is sufficient to gain the 

 writer's object. 



Spherical Shell or Bowl or Dome. 



Consider the shell to be supported on a horizontal plane, 

 all the circumstances being symmetrical about the vertical 

 axis. Let the elements of stress at any point be (P, Q, R, S, 

 Tsin# and U sin 0), and the internal and external radii, a 

 and b. Then the equation connecting S and T is an inde- 

 pendent equation and, for our purpose, we may assume S and 

 T to vanish. 



The remaining equations are 



-§+2P-Q-R+2U*-(l-^)g=^, • • (!) 



where as stands for cos 6. 



Without indicating here the reasons for the step, let us 

 assume 



Q=U» +W r£=4 + ?, (3) 



jl io r 



R = Utf-(l-* 2 ) — -gprx-gpr^^ + - , . . (4) 



where the material ends with #=a in case the shell is an open 

 bowl. Otherwise a=l. 



* Communicated by the Author. 



