24 Dr. C. E. Weather burn on Two Fundamental 



continuous at the boundary. For it is to a constant multiple 

 equal to 



2 i I v(Y) . ixlnx grad ~\ Ids = I v(s) x j grad( - J X n Ids. 



It is clear from this form of the expression that w 2 (p) is 

 continuous at the boundary. For as p moves up to t along 



the normal and coincides with it, grad -eft remains a finite 



r /l 



vector in the direction of the normal, so that grad (-)xndt 



vanishes in the limit. Hence there is no discontinuity due 

 to the element dt, and w 2 (p) is continuous at the boundary. 

 Lastly, the integral w 3 (p) arising from the third term of 

 (7) is, as in § 4, given by 



W ^> = i IT* J Y W • [ grad r gl ' ad *"] Tn (," ) *' 



This is an integral with discontinuity at the boundary equal 

 to one-third of that of the corresponding ordinary double 

 stratum * ; so that 



Ws(i+)= r+£ T (o+w,(o 



( r )=-TTi T(0+W3(<) 



Combining tbe results for the three potentials ^ (p), vr^ipY 

 and w 3 (p) we have the discontinuity for W(p) expressed 

 by (14). . . 



The continuity of the normal derivative of an ordinarj'- 

 -double stratum potential has its counterpart in a further 

 property of the vector potential WQe), viz. that if the func- 

 tion v (.?) is finite and continuous along with its first derivative, 

 the surface traction due to a displacement of the particles 

 represented by W(p) is continuous at 2. This may be put 

 more definitely as follows. Imagine a surface 2' close to 2, 

 and with its normal everywhere parallel to the normal to 2. 

 Then the surface traction due to the displacement W(p) of 

 the body bounded by 2' remains continuous as 2' moves up 

 to and passes through 2. This theorem, which is due to 

 Lauricella f, is true when v (t) is finite and continuous along 

 with its first derivative. 



* Cf. Lauricella, loc. cit. pp. 161-164. 



+ Cf. Atti Lincei (5), t, 15 (1906), p. 429 ; also AnnalidiMat. (1907), 

 €ap. II. § 6. 



