492 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



plate, and to be held in its actual position, partly by the forces directly applied to its 

 mass, and partly by the action of the remainder of the plate exerted across the 

 boundary ; if the plate be now cut out, it will be necessary, in order to hold it in the 

 same configuration, to apply at every point of its edge a distribution of force and 

 couple identical with that exerted by the remainder before the plate was cut out. 

 Now, it has been shown by KIKCHHOFF * that these equations express too much, and 

 that it is not generally possible to satisfy them ; but the method proposed by 

 THOMSON and TAIT! gives a rational explanation of KIRCHHOFF'S union of two of 

 POISSON'S boundary-conditions in one, and renders his theory complete. However, 

 the objection raised by DE ST. VENANT| to the fundamental assumption that the 

 stresses and strains in an element can be expanded in integral powers of the distance 

 from the middle-surface, seems to require a different theory. 



The next epoch in the theory of plates is marked by KIRCHHOFF'S memoir just 

 referred to. The method rests on two assumptions, viz. : (1) Every straight line of 

 the plate which was originally perpendicular to the plane bounding surfaces remains 

 straight after the deformation, and perpendicular to the surfaces which were originally 

 parallel to the plane bounding surfaces ; (2) all the elements of the middle- surface 

 (i.e., the surface which in the natural state was midway between the plane parallel 

 bounding surfaces) remain unstretched. Both these assumptions may be shown to be 

 approximately true in the cases of flexure and transverse vibration, but, as assump- 

 tions, they appear unwarrantable. In this memoir of KIRCHHOFF'S the union of two 

 of POISSON'S boundary-conditions in one was first effected, the method employed to 

 obtain the equations being that of virtual work. The theory of this memoir will be 

 referred to as KIRCHHOFF'S " first theory." 



KIRCHHOFF has given a general method for the treatment of elastic bodies, some 

 of whose dimensions are indefinitely small in comparison with others. In this method 

 we consider, in the first place, the equilibrium of an element of the body all whose 

 dimensions are of the same order as the indefinitely small dimensions. When we 

 know the potential energy due to the internal strain of such an element, we obtain 

 by integration over 1 the remaining dimensions the whole potential energy due to the 

 elastic strain of the body. Then, taking into account all the forces which act on the 

 body, we can form the equation of virtual work, which will lead directly to the 

 differential equations and boundary-conditions of our problem. 



In KIRCHHOFF'S method it appears that, to a first approximation, the bodily forces 

 produce displacements which are negligible compared with those produced by the 

 surface-tractions exerted upon the element by contiguous elements, and that, to the 



* "Ueberdas Gleichgewicht und die Bewegung einer elastischen Scbeibe," ' CBELLK, .Tourn. Math.,' 

 vol. 40. 



t Loc. cit., pp. 190-1. 



I Translation of CLEBSCH'S 'Elasticitat,' Note on 73, p. 725. 



' Vorlesungen Uber Mathematische Physik,' pp. 406 et seq. 



