AND DEFORMATION OF A THIN KLASTIC SHELL. 493 



same order of approximation, the displacements, when divided by finite quantities of 

 one dimension in length, are negligible compared with the strains. 



The application of this method to the theory of plates nppears to have been first 

 made by GEURING, a pupil of KIRCUHOFF'S, at the latter *s request ; and the results 

 will be found in Kii:< HIIOKF'S thirtieth lecture, and in CLEBSCH'S ' Theorie der 

 Elasticitiit fester Ktirper,' (54 et seq. We shall call the theory thus deduced KIRCH- 

 HOFF'S "second theory." POISSON and KIRCHHOFF had both arrived at the equations 

 S, T, R = 0,* which express that the traction exerted on an element of a surface 

 normal to the middle-surface of the plate is everywhere tangential to the middle- 

 surface. These equations are fundamental in KIRCHHOFF'S second theory. This 

 appears to lie at the root of the objection raised by DE ST. VfiNANxt to this theory, as 

 it is stated by him that S and T, if they exist, may produce important effects, 

 especially when the material of the plate is not isotropic. 



It seems unnecessary to explain in detail THOMSON and TAIT'S treatment of the 

 problem. We need only note here that the equations S, T, II = are a basis for this 

 theory also. 



\_AddedJuly, 1888. An important inference from the method is that a line of 

 particles initially normal to the middle-surface is approximately normal to this surface 

 after strain. This is expressed by the vanishing of the shears a and b, as given by 

 equations (11) infra. This conclusion is intimately bound up with the conclusion that 

 S and T vanish. At the edge of the plate S and T may have given values which do 

 not vanish, and the approximate perpendicularity of line-elements originally perpen- 

 dicular to the middle-surface will here break down. The transition from a state of 

 things in which S and T exist at the edge to one in which they vanish, on a surface 

 parallel to the edge and very near to it, is illustrated by the discussion in THOMSON 

 and TAIT'S 'Natural Philosophy,' 721-729. The conclusion seems to be that 

 KIRCHHOFF'S general method for the treatment of elastic bodies, some of whose 

 dimensions are indefinitely small in comparison with others, cannot be applied to the 

 elements. situated very near to the edge of a plate, as the strain is not produced in 

 these by the action of contiguous elements. We may, nevertheless, regard it as 

 giving correctly, not only the potential energy due to the strain of an element at a 

 distance from the edge, but also the whole potential energy arising from the strain in 

 all the elements. It will thus lead us to the right differential equations of motion ,or 

 equilibrium and boundary-conditions.] 



The theory of the flexure of an elastic plate has been placed in a much clearer light 

 by the researches of BOUSSINESQ, who has treated the subject in a masterly manner 

 in two memoirs. In the first of thesej he has certainly proved that S = 0, T = 0, 

 R = is an approximation to the actual state of stress within an element of the 



* I use THOMSON and TAIT'S notation for the stresses, strains, and elastic constant*. 

 t Translation of CLKBSCH'S ' Elasticitat,' p. 691. 

 J ' LIOUVILLE, Journal do Math.,' 1871. 



