66 



Dr Ghree, The Equilibrium of Isotropic [Jan. 27, 



u=- 



b s p' 



a 3 — b s \ 3m — n 



a s p r ( ab) 3 (p' 



■P) 



4>nr 2 



.(17). 



a? ( _ (2i 2 + 2i-S)m + n 

 + e h <n{P P) 4n ( 3m _ w ) ' 



„ 1^1 = 91 , l( 1 ^A _ea 2 p-p' 

 I old I\sm6 d<f>)~ h 4>n 



The subsidiary terms vanish when jp' = p. It is obvious in fact 

 from (11) to (14) that they must do so in any case, irrespective of 

 the magnitude of (a — b)/a. Along the same radius vector they 

 are constant, to the present degree of approximation. The varia- 

 tion of the principal term in u along a radius vector was considered 

 in my previous paper (see (13^) &c. writing — p for R, — p for R). 

 This variation is important only when (p ~ p')/p is small, and in 

 this case the subsidiary terms in e tend to vanish. As our present 

 object is to examine the significance of these terms it will be 

 supposed that (p ~ p')jp is not small. Omitting the small part of 

 the principal term in u which varies throughout the thickness, we 

 may write (17) in the form 



u= — 



<#{p-p) 



h 



da, 

 d6 



= w 



4*n E 

 da 



eai {(2i 2 + 2i - 3) m + n] 



4>n (3m — n) 

 _ ea 2 (p — p) 

 sin 6 d(j) J 4<nh 



.(18), 



where E is Young's Modulus, rj Poisson's ratio. 



Remembering the values of the direction cosines of the normal, 

 we see that (18) is equivalent to the following three displacements : 



a displacement — a 2 (p — p')/4mh along the normal, 



„ „ 7)a 2 (p—p')/Eh „ „ radius vector, 



ea { a 2 (p - p) {(2i 2 + 2i-S)m + n] 



4<n (3m — n) h 



along the normal 

 or radius vector. 



Thus to the present degree of approximation the displacement 

 is along the normal, that is in the line of action of the applied 

 force, when 



?7=0. 



The nature of the change produced by the pressures in the 

 shape of the thin shell is easily found as follows. It will suffice to 

 treat the outer surface (1), as the results for the inner are of the 

 same character. Denote the radius vector by p, i.e. let 



eo- ? - = (p- a) fa. 



