22 



Dr. C. E. Weatherburn on Two Fundamental 



the region in terms of the boundary values of the function 

 and those of its normal derivative. But this expresses the 

 harmonic function as the difference of two others which are 

 potentials of double and simple strata respectively. It is- 

 therefore an obvious suggestion to examine the two integrals 

 in (11) from the point of view of stratum potentials. This 

 has been done by Lauricella * for the case in which the first 

 integral is modified by using only the self-conjugate part of 

 its dyadic. The elimination of the anti-self-conjugate part 

 is the object for which his idea of pseudo-tension is intro- 

 duced. We shall show, as Lauricella did in the modified 

 case, that the two integrals in (11) possess properties and 

 boundary discontinuities analogous to those enjoyed by 

 the Newtonian and logarithmic potentials of double and 

 simple strata. 



§ 5. First, it is evident that each of these integrals satisfies 

 the equation (1) which characterizes a displacement. For 

 each of the vectors s 1 (ps), s 2 (ps), s 3 (ps) as a function of p 

 is a solution of that equation, and so also is D{p) by hypo- 

 thesis. Hence the first integral of the second member of 

 (11) must satisfy (1). But as a function of p this is inde- 

 pendent of the boundary function D(s) ; hence the integral 

 is a solution of (1) whatever be the finite and continuous 

 function D(«). Indeed it is easily verified that the function 

 F (sp) satisfies (1) for all values of a. If for brevity we 

 denote the dyadics in (11) by the symbols f 



*(<P) = ^ [*i(*p) i+ F 2 (*p) J + F 3 (sp) k] 



®(pq) = 9Z t is i (pq) + J s 2(pq) + k s 3 (p?)] 



, . . (12) 



then the integrals 



W(p) = jv(*) .W(sp)ds 



V(p) =§®(ps).u(s)ds 

 in which u(s) and v(s) are finite and continuous vector 



(13) 



* II Nuovo Cimento, loc. cit. pp. 155-174. 



f I have chosen the notation to resemble that employed in my earlier 

 papers on the potential theory. Cf. Quarterly Journ. vol. xlvi. (191-4- 

 15). Three papers. 



