230 Mr. A. B. Basset on the Difficulties of Constructing 



expression is given by (13), which contains the three unknown 

 quantities A, A 1? and -sr. 



10. Although the existence of external pressures renders 

 it inadmissible to trent R as zero, yet the stresses S and T 

 still vanish at the surfaces of the shell. I have shown in my 

 papers that the terms of lowest order which these stresses 

 contain are quadratic functions of h, h' ; and the arguments 

 by which this result was obtained are unaffected by the 

 existence of external pressures. Since the problem we are 

 discussing is one of two dimensions, T will be accurately 

 zero ; also since 



N=f* SW, 



and S' is a quadratic function of h and h', it follows that N 

 must be proportional to A 3 , whence by the third of (6), 

 dQjds is proportional to h 3 . 



11. We have now to consider how A and A 1 depend upon h. 

 Since the quantity A is the value of R at the middle 



surface of the shell, it cannot contain any negative powers 

 of A, otherwise R would increase indefinitely as the thickness 

 of the shell diminishes indefinitely, which is impossible, since 

 R must lie between — II j and — n 2 . It is, however, possible 

 for A to contain a term independent of the displacements ; 

 but such a term need not be considered, inasmuch as it 

 disappears on differentiation. 



The quantity A 1 might, however, contain a term involving 

 A -1 ; for the value of R at a point near the middle surface is 



A + AJJ+ ... 



When the thickness of the shell diminishes indefinitely, h 1 

 approaches the limit h, but in such a manner that h! is always 

 less than h ; if therefore A x contained a term involving hr 1 , 

 the quantity A^J would not become infinite in the limit. 

 On the other hand, if A 1 did contain such a term it would 

 necessarily be independent of the displacements, inasmuch as 

 the term in question must disappear on differentiation, 

 otherwise dGc/ds would not be proportional to h 3 . 



It therefore follows that the only portions of A, Aj which 

 it is necessary to consider are those portions which are 

 independent of h. I regret to say that I have not succeeded 

 in discovering any method by which their values may be 

 rigorously deduced ; but although as a rule I distrust argu- 

 ments founded on general reasoning, yet the difficulty of 

 obtaining the values of these quantities by a rigorous mathe- 

 matical investigation having hitherto proved insuperable, 



