38 Two Fundamental Problems in Theory of Elasticity. 



orthogonal to each o£ these six vectors. The six conditions 

 thii3 expressed are equivalent to the two relations 



§t(t)dt = 0, §p(t)xf(t)dt = Q, . . . (63) 



which are the conditions of equilibrium of the body S acted 

 on by the surface forces f(t). These relations must a priori 

 be satisfied if the problem is to admit a solution. Then (60) 

 admits a finite and continuous solution u(t) which, as density 

 of a simple stratum, defines a potential Y(p) representing 

 the required displacement. This displacement is, by (31), 

 equal to 



v<» = j'r r +1 (p<) .t{t)dt^^v +1 (pt + ) .t{t)dt. (64) 



§ 16. If the body occupies the infinite outer region S', the 

 parameter value for the problem is X= — 1, and the dis- 

 placement of the body for a given surface traction — f (t) is 

 expressible as the potential of a simple stratum of density 

 u(s) given by the integral equation 



u(t) + §VL(s).V(ts)ds = f(t). . . . (65) 



Now the corresponding homogeneous equation 



u(t) + §u(s).'V(ts)d8 = .... (66) 



does not admit any solution but zero. For suppose that it 

 admits a solution Ui(t) ; then the simple stratum displacement 

 Vi(p) with this function as density has zero surface traction 

 at the boundary of the outer region. The function V\(p) 

 for the region S' must therefore represent one of the 

 degenerate displacements ; and since it vanishes at infinity 

 this displacement is identically zero throughout S'. But the 

 function is continuous at 2, being the potential of a simple 

 stratum. Hence it vanishes over the boundary of the inner 

 region, and therefore also throughout that region. The 

 surface traction at the point t + is therefore zero. Thus 



2u 1 (*)=F(r)-F(> + )=0, 



which proves the statement. 



Since then (66) does not admit any solution but zero, 

 (65) does admit a unique finite and continuous solution u(s) ; 

 and this function substituted in (26) gives the required 

 solution of the problem for the outer region S', which by 

 (31) may be expressed in the form 



