Problems in the Theory of Elasticity. 23 



functions of the boundary point s, are solutions of the 

 equation (1) of elastic equilibrium, and are finite and con- 

 tinuous vector functions of the point p of the region bounded 

 by 2. The function W(p), whose dyadic V(sp) involves 



i I - I, will be spoken of as the vector potential of a double 

 an\r / 



elastic stratum of moment v(s) ; while VQo), whose dyadic 



is similarly related to -, will be called the vector potential 



of a simple elastic stratum of density u(s). It will be 

 observed that the density and moment as so defined are 

 vector functions of the position of the point s of the boundary. 

 When no ambiguity is possible these potentials will be called 

 briefly potentials of double and simple strata respectively. 



§ 6. Vector Potential of a Double Elastic Stratum. — The 

 double stratum vector potential W (p) defined in the previous 

 section has discontinuities at the boundary exactly resem- 

 bling those of the ordinary double stratum potential, and 

 eKpressible in the form 



W (t + ) = t(*) + j v (s) . \F (st) ds 

 W(r) = -v(*)+ Jv(» .y(st)ds 



(14) 



t + being a point of the inner region indefinitely close to the 

 boundary point t but not on the boundary, and t~ the corre- 

 sponding point of the outer region. To prove this we shall 

 consider separately the parts of W (p) due to the three terms 

 of F (s) given by (7). First the potential 



!»(! + *) J 



fait) 



Is 



is, to a constant factor, that of an ordinary double stratum 

 of moment v(s). This satisfies the known relations 





wi(r 



i + k 

 l 



v(0+ w i(*)" 



(15) 



wi(r)=.j-^v(0+wiWJ 



The potential w 2 (p) arising from the second term in (7) is 



