p 



26 Dr. C. E. Weatherburn on Two Fundamental 



to the displacement V(p) is given by 



P W = ^ \ [iF/(^)+jF 2 '(^) +kP 3 '(^)j.up)^ 



•J 



= (Va). ¥'(*&) A (20) 



the dyadic ^'(sS) being derived from ^(sd) replacing d/dn 

 by d/dn'. This function F(«?) resembles the double stratum 

 potential W(p) of the previous section, but the dyadic in 

 (20) differs from that in the expression for W(s) in the order 

 of the variables. The effect of this upon the boundary dis- 

 continuity of the expression appears thus. As X' moves up 

 to and coincides with 2, s moves up to and coincides with a 

 point t. But whether s is the point t + of the inner region 

 or the point t~ of the outer region, the value of the ex- 

 pression F/(s£) given by (19) is equal to —F 1 '(ts) = —'F [ (ts) 

 given by (7) ; for the change in the order of the variables- 

 means a change in the direction of r. But it is to the 

 element dt in the integration that the discontinuity is due. 

 Hence the discontinuity in the function (20) is opposite in 

 sign to that of W(p). If, then, F(£+) and F(r) denote the- 

 surface tractions for the inner and outer regions respectively 

 due to the simple stratum displacement V(jt>), 



F(£+) = -u(f) -f Ju(3) . ¥(*$) d$*) 



F(r) = u(t) + Ju($) .V(t$)d$y 



It should be observed in what relation the formulae (21) 

 stand to (14). Regarded as equations in v(S) and u(S) the 

 first of (14) and the second of (21) are integral equations ; 

 but they are not associated. Though the order of the vari- 

 ables is different in the two kernels, the unknown occurs as 

 a prefactor in each case. The equation associated to (14 a) 

 is obtainable from (21 b) by making the kernel the prefactor. 

 The difference is vital because the dyadic is not self-con- 

 jugate. Its value is neatly expressed by the formula 



2 7 rCl + ^ ) ^(^)=I^(- 1 )-Ix[o- r adQx 1 x] 



+ 3&(gradrgradr)^Q. 



The first and last terms in this expression are self-conjugate 

 dyadics, but the second is anti-self-conjugate. 



