Problems in the Theory of Elasticity. 



§ 7. Vector Potential of a Simple Elastic Strat 

 sider next the simple stratum potential 



25 



urn. — Con- 



V 0) = 4r~| (" iSl ^0+JM/w)+ kB 8G«) J •»(')* 

 i 



<t>(ps).xi(s)ds (17) 



The dyadic O (ts) is conjugo-symmetric, being both sym- 

 metric and self-conjugate. It is symmetric because the 

 function s (ts) is symmetric. It is also self-conjugate ; for 

 if expressed in nonion form it has the same coefficient for 



k ~d 2 r 



i j as for j i, viz. — . " ^s~ • The dyadic is therefore 



conjugo-symmetric and enjoys all the properties established 

 for such a kernel in the last part of my paper first 

 referred to. 



We observe that the function V{p) is continuous at the 

 boundary. This is clear from the form of s (ts) which 

 becomes infinite at t = s only like 1/r. Further, V(p) satis- 

 fies (1) and may therefore be regarded as the displacement 

 of the point p of a body occupying the region bounded by 

 some surface to be specified. The corresponding surface 

 traction varies with this surface, and will now be shown to 

 be discontinuous * at 2. Consider a surface 2' , either within 

 or without 2, with its normal n' everywhere parallel to the 

 normal n to 2. Let s, cr denote points on %' ; t, $ on 2. 

 Because the function s (pq) is symmetric we have 



•,(#»)=■!(»«),. ..... (18) 



representing a displacement at th^ point s with $ as the 

 pole. The surface traction on 2' due to this displacement 

 is therefore F/(.s3), whose value is obtained from (7) re- 

 placing djdn by d/dn', so that 



^^=lii(')-TT^ ix ( a ' Xsrad J) 



+ 



U d 

 1 + k dn 



>(l)(%>^> • • ^ 



the point s ( = f, rj, f) being the current point and d the pole 

 from which r is measured. Hence, since in (17) u(3) is 

 independent of s, the surface traction at the point s of 2' due 



* Cf. Laurieella, II Nuovo Cimento, loc. cit. pp. 166-174. 



