AND DEFORMATION OP A THIN ELASTIC SHELL. 503 



In (6) and (7) suppose u' = , v = r , w' = w ; then 



? = p L () + ?! (!,) + 4 r> th five other 



subtracting these from (6) and (7), we find 



A 





A,A, 3y ~ A, A,*, 



/o\ 



4. These are simply the conditions of continuity of the mass of the shell when 

 deformed. To obtain the forms of u lt v lt w l from them we shall have to introduce 

 stress-conditions. As the quantities in (9) are small, it will be sufficient to omit 

 products, and so form equations of equilibrium of the element referred to the 

 orthogonal coordinates ( p, q, r) as if we were referring to fixed axes at P. 



If A, B, C be three functions of r to be determined, we have 





Ilence, for the six components of strain, and for the cubical dilatation 8, 



A>i / tCa f C/\_/ 



To determine A, B, C, we have the stress-equaticns 



