MancJicstcy Memoirs, ]^oI. h. (191 1 ), No. '/JO. 7 



and in the resulting equations I .shall replace cosf^ by x. 



I shall also assume symmetry about the vertical diameter. 



Then, if the stresses are supposed continuous about 



tiie point under consideration, we shall have the equations 



cr dx 



cr I - X ex 



r{^+3r-l^+2S-^^..=^0 (i) 



cr ex I - .V" 



It is clear that 6" and T do not appear in the first two 

 of these equations, and that we are at liberty to proceed 

 with those two only. To propose that .S" = 0, T—0, may 

 not be unreasonable ; this would require that the tangent 

 to a parallel of latitude at any point in the shell should be 

 a principal axis of the stress quadric at the point, but it is 

 not very material to the procedure at the moment. 



For the purpose of engineers, and for practical 

 purposes generally, we have to deal with structural rather 

 than molecular forces. I, therefore, multiply each of these 

 equations by /v/r and integrate from r = a at the interior 

 surface of the shell to r=d, at the exterior surface, noting 

 that P = 0, 7'=0, 6^=0 at each such surface. 



We thus obtain, writing Q for I Ordr, R for I Rrdr, 



f7 for I Uuir, S for I i)';v/;', and 7' for 1 Trdr, and inte- 

 grating, 



Q = Ux - -—- + ( I - o)«'^-l-,, 

 \+x I - a:- 



r, f-r , "^G , w I s I 



R = f/.v - ( I - x-)^^- - wx + - ( I - ii)w- 



t.v I + .,v I — .V" 



(i--v-)r=i{(i-.v-^)6l, (2) 



ex 



