34 Dr. C. E. Weatherburn on Two Fundamental 



Reverting to the equations (33) we see that the first 

 integral, regarded as a function of p> is of the nature of a 

 simple elastic stratum potential of density XR'(tq). From 

 this it follows that 



l[Tr'o + 5 )-Tr'(r ? )]=xH'(< ? ), 



-i[Tr'(t + y) + Tr'(ry)]=x(t ? ) + xJ X (««).H'(«7)rf* 

 Adding and subtracting, we find 



Tr'(r?) = -(l + x)H%),| 



TF(*+2)=-(l-\)H'(ty).J * * ' V / 



An exactly similar pair of equations (50) with undashed 

 letters may be deduced from the first form of (33') • From 

 the second integral in (33') which, regarded as a function 

 of q, is of the form of a double stratum potential of moment 

 \T(ps), we find in a similar manner 



Fo*+)=(i+\)ron r , n 



rcpr)=(i-\)r( P or 



showing the nature of the boundary discontinuity of the 

 dyadic T(pq), which I shall show elsewhere to be the 

 Green's dyadic for the inner region corresponding to zero 

 surface displacement. A similar pair of relations (51') hold 

 for the dyadic Y'(pq). From the preceding results and the 

 equation (47) the following relations may be deduced * : — 



TG'(t-p)=-(l + \)K'(tp) + V'(tp)\ , 91 



TG'(«»=-(1-\)K'(^)-P'(^)f " ' V " ] 

 and 



G(g*+) = (1+X)G( 2 0-VK?01 f53) 



Gr( q t-)={l-\)Gc{qt) + \ Q t (gt)} K } 



§ 13. The First Boundary Problem for one Region only. — 

 Consider now the boundary problems for the parameter 

 values X— + 1, which correspond to the inner and outer 

 regions separately. Taking the first boundary problem for 

 the inner region S, (X= — 1), the moment v(s) of the double 

 stratum whose vector potential solves the problem is given 

 by the integral equation 



v{t) + §v(s).yr(8t)ds = l(t). . . .• (54) 



To prove that \= — 1 is not singular for the kernel "^(ts), 



* Ibid., § 2. 



