Problems in the Theory of Elasticity. 27 



§8. The First and Second Boundary Problems. — T shall 

 consider the fundamental problems of elastic equilibrium in> 

 a general form analogous to that proposed by Poincare for 

 Dirichlet's and Neumann's problems, an arbitrary parameter 

 X being introduced into the prescribed boundary conditions. 

 We set before ourselves the determination of displacements 

 W(p) and V{p) for an elastic inotropic body corresponding 

 respectively to the boundary relations 



i[W(«+)-W(r)]-^[w(« + )+W(r)]= f(«)l 

 i[TV(r)-TV(« + )]-5[TV(r) + TV(< + )]=-f(<)( 



(25) 



f(t) being a given piecemeal continuous function of the 

 boundary point, and TV(t + ) denoting the surface traction at 

 the point t due to the displacement V(p) for a body occupy- 

 ing the inner region, while TV(t~) has a similar meaning 

 for a body occupying the outer region. The problems (25)> 

 will be called the first and second boundary problems re- 

 spectively. For the parameter values X= ±1 they relate to 

 the inner and outer regions separately. 



Endeavouring to satisfy these by vector potentials 



W(p) = fv(s) .V(sp) ds~) 



. [, ... (zsy 



Y(p)=]®(ps).TL(s)ds) 



due respectively to a double stratum of moment v(s) and a 

 simple one of density 11(5), we find from the boundary 

 properties of such that V{s) and u(s) are solutions of a pair 

 of vector integral equations 



T(0-XjT(»).*(jt)l& = f(t)-l 



u(0-Xjx(««) •»(«)*=*(■*) J" 



where 



that is the dyadic conjugate to W(ts). These integral equa- 

 tions are not associated because *fr(ts) is not self-conjugate. 

 The kernels of the equations (27), 



^ (**)== ^ [F,(««)i + P a (««) J + J , ,(««)i:]' 



and its conjugate, become infinite at the point s = t but of 

 order less than two, and therefore Fredholm's method of 



