[ 15 ] 



III. On Two Fundamental Problems in the Theory of 

 Elasticity. By C. E. Weatherburn, M.A. (Cantab.), 

 D.Sc. {Sydney) ; Lecturer in Mathematics arul Physics, 

 Ormond College, University of Melbourne (Australia)*. 



§ 1. Introduction. 



TI^HE theory of vector integral equations, which I have 

 A treated elsewhere f, finds an important and direct 

 application in the fundamental problems of elastic equi- 

 librium, requiring the determination of the displacement at 

 any point of an elastic body when the value of the surface 

 displacement is known or that of the surface traction. 

 These will be referred to as the first and second boundary 

 problems respectively. In the present paper I generalize 

 the problems by the introduction of a parameter X, in the 

 manner proposed by Poincare J in his discussion of the 

 problems of the potential theory. Moreover, by the use of 

 vector analysis I construct, from Somigliana's integrals 

 of the equations of equilibrium, dyadics which form the 

 basis of displacement functions whose properties exactly 

 resemble those of ordinary simple and double stratum poten- 

 tials. The treatment of the fundamental problems in elas- 

 ticity is thus recast, and will be found to run exactly parallel 

 with that of the potential problems. Corresponding theorems 

 are established concerning the magnitude and the reality of 

 the singular parameter values, and the simplicity of the pole 

 of the solution at each of these. The singular case of the 

 potential problem for the inner region when the value of 

 the normal derivative is known, finds its counterpart in the 

 problem of equilibrium of an elastic body under given 

 •surface traction. 



The integral equations that arise are vector equations with 

 dyadic kernels and dyadic resolvents. Connected with the 

 resolvent of the dyadic which forms the basis of a double 

 stratum displacement is another, which, in a separate paper §, 

 I prove to be a " Green's dyadic," analogous to the Green's 

 function for Laplace's equation vanishing over the boundarv. 



Lauricella || was led, in order to escape the difficulty of 



* Communicated by the Author. 



"t " Vector Integral Equations and Gibbs' Dvadics," Trans. Camb. 

 Phil. Soc. vol. xxii. pp. 133-158 (1916). 



X Cf. "La me'thode do Neumann et le probleme de Dirichlet." Acta 

 Math. Bd. 20 (1896). 



§ " Green's Dyadics in the Theory of Elasticity," Proc. Lond. Math. 

 Soc. 1916-17. 



|| Atti M. Ace. Lincei (5), t. 15 2 , pp. 75-83 (1906); also II Nuovo 

 Cimento (5), t. 13 (1907). Four notes. 



