62 Dr Chree, The Equilibrium of Isotropic [Jan. 27, 



in detail only with a simple case of the shell problem, viz. that in 

 which the forces consist exclusively of uniform normal pressures, 

 different over the two surfaces. 



The equations to be solved prove to be ultimately identical 

 with those treated by me in the Society's Transactions, Vol. XV., 

 p. 351. Attention is mainly directed to the case of a thin shell 

 whose surfaces are concentric, similar and similarly situated, as 

 being that of most physical interest. 



Let p, p denote the uniform pressures, and let the equations 

 to the two surfaces be 



r = a(l + eo-i) (1), 



r = b(l + e'a i ) (2), 



where e and e' are small quantities whose squares are neglected, 

 and <Ji is a surface harmonic of degree i. For simplicity there is 

 supposed to be only one harmonic term, but the method obviously 

 applies, however numerous these terms may be, supposing the sum 

 of their numerical values small compared to unity. 



Referred to the fundamental polar directions r, 6, <f> at any 

 point on the surface (1), the direction cosines of the normal, since 

 e 2 is neglected, are 



d<Ti 1 dcri 



' ~ 6 dd' ~ 6 sin~<9d0 ; 



similar results with e' written for e apply over (2). 



If the shell were truly spherical the complete solution would 

 consist in the radial displacement 



[b s p'-a 3 p r | (ab) 3 p'-p) ^* 



a z — b z ( 3m — n r 2 4>n 



where m and n are the elastic constants in the notation of 

 Thomson and Tait's Natural Philosophy. 



Answering to (3), the dilatation A and the stresses, in the 

 notation of Todhunter and Pearson's History, are given by 



A = (b 3 p' - a s p) + i(™-|) 3 - & 3 )| , 

 ™ = ^ITp W ~ a3 P + (jr) (P ~ P') 



™ = H> = a^ 3 { b3 P' -* 3 P + l (tJ (P' ~ 

 r~0=r<j) =00 = 



* Cf. Transactions, Vol. xv., p. 343. 



l...(4)' 



