Problems in the Theory of Elasticity. 37 



where a and <w are constant vectors. This proves the state- 

 ment. The solution W(/>) to this problem is given by the 

 double stratum potential 



▼d»)-j*(0-K+i«p)*, 



which, in virtue of (52), may be expressed in the alternative 

 form 



W(p)=-ijf(0.TG +1 (O>>ft. 



§ 15. Second Boundary Problem for one Region oidy. — 

 Consider next the second boundary problem requiring the 

 determination of the displacement for a given value of the 

 surface traction. In the case of the inner region the particular 

 value of the parameter is X= + 1, and the required displace- 

 ment for a given surface traction f (t) is expressible as the 

 potential of a simple elastic stratum of density m(s) given by 



n(0-Jx(«»)-»W^=*(0- • • • ( G °) 



The value \=+l is, however, a characteristic number of 

 the kernel X(ts) ; for the homogeneous equation 



u(0 = J*(k) •*(»>*» .... (61) 

 and its associated 



u(t) = §TL(s).x(st)ds = §y{st).u{s)ds . . (62) 



admit certain non-zero solutions. The last equation may be 

 written 



and that this is satisfied by any constant vector a is easily 

 verified by considering separately the three parts Wi(t), 

 w 2 (0, and w 3 (0 as in § 6. We thus find 



Wl (0 = a/(1 + Z;), wj(0=0, w 3 (0 = /ca/(l + &), 



showing that a is a solution of (62). Further, if p is the 

 position vector of the point p relative to the cm. of the body S, 

 and the unit vectors i, j, k be taken along the principal axes 

 of inertia, it can be similarly shown that ix/a, jX/o, k x /> are 

 also solutions of (62). The only independent solutions of 

 this equation are the six vectors i, j, k, ix p s j x p, k x p. 



Hence, in order that (60) may admit a finite and con- 

 tinuous solution, it is necessary and sufficient that f (t) be 



