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



general equations of an elastic solid on this hypothesis. Under 

 any circumstances, an assumption of this character is an 

 exceedingly objectionable one to found a theory of thin plates 

 upon ; and Clebsch himself appears to have been conscious of 

 the imperfections of his theory, for he says (Theorie de V Elas- 

 ticity, Saint Venant's translation, p. 296): — " Ces equations 

 (i. e. R = S = T = 0) ne s'appliquent tout d'abord qu'aux 

 valeurs de z correspondant aux bases ou faces extremes de 

 la plaque : mais j'examinerai seulement les etats d'equilibre 

 dans lesquels elles sont satisfaites en tons les points du corps" 

 The assumption itself can, however, be shown to be erroneous; 

 for if a plane plate of thickness 2A is slightly bent into a 

 cylinder of any form, and N be the normal shearing-stress 

 across, and G the flexural couple about a generator, one of 

 the equations of equilibrium is 



^+N=0. 

 ds 



But 



\ h Hd 



£ 



where h 1 is the distance of any point of the substance of the 

 shell from the middle surface ; if, therefore, T were zero, 

 N would be zero and G constant, which is wrong, since G 

 is known to depend upon the change of curvature. Similar 

 observations apply to S. The fact that two of Clebsch's fun- 

 damental assumptions can thus be proved to be erroneous, 

 vitiates every theory in which they form an essential part of 

 the investigation. 



3. But notwithstanding the defective character of the 

 theories referred to, Clebsch's assumptions, when properly em- 

 ployed, are capable of furnishing a perfectly satisfactory theory 

 of thin plates and shells, subject to the following limitations : — 



(i) The surfaces of the plate or shell must not be subjected to 

 any external pressures or tangential stresses. 



(ii) The results of the theory must be regarded as approxi- 

 mate ones, in which powers of the thickness higher than the cube 

 are neglected. 



The reason of this is that K, S, and T are each of the form 



uh 2 + 2/3hh' + yh' 2 + terms involving higher powers of h, h', 



w 7 here a, /3, y are functions of the displacements of the 

 middle surface and their differential coefficients ; whence it 

 can be shown that if these stresses were retained, they would 

 introduce into the expression for the energy of the system 

 higher terms than h B . Since such terms are to be neglected, 



