20 Dr. C. E. Weatherburn on Two Fundamental 



§ 1. Vector Potentials of Elastic Strata. — If D and D' are* 

 two solutions of the equation (1) regular within the given 

 region, and F, F' the corresponding surface tractions, the 

 bodily forces being supposed zero, Betti's reciprocity theorem* 

 may be put in the form 



§¥(s).n'(s)ds-§T'(s).V(s)ds = 0. ... (8) 



Take the usual set i, j, k, of three unit rectangular vectors, 

 and consider the values of the particular integral s of the 

 previous section when the vector a is replaced successively 

 by i, j, k. Denote its values in these cases by s 1? s 2 , s 3 , and 

 the corresponding surface tractions by F^sp), F 2 (sp), ^^( S P) 

 respectively. Then 



l 

 s i = z 



2(1 + *) 



k 



graddiv(ir), I 



= '--27ITI)S raddiv(j ^ • • • (9) 



grad div (kr). 



2(1 + *) 



We may in the formula (8) replace D' by s b s 2 , s 3 in suc- 

 cession provided the point p be isolated by (say) a small 

 sphere Z with p as centre. It then follows that 



( V(s).s m (ps)ds = £ F m (sp).B(s)ds 1 (m = l,2,3). 



Js+z Js+z 



Multiplying these in order by i, j,k, and adding, we find a 

 result which may be written 



J 



[isi(ps)+js 2 (_p)+ks 3 (_p)].F(>)rfs 



2+Z 



= f D( 5 ).[F l (^)i + F 2 ( 5 p)j + F 3 ( 5J 9)k]^, . . (10) 



J2+Z 



where the expressions in square brackets are dyadics. In 

 evaluating the integrals over the small sphere Z, we notice 

 that the first member contributes nothing ; for s 1? s 2 , s 3 

 become infinite at p only of the first order, while the area of 

 the small sphere is of the second order. Considering the 

 second member, we take the values of F l7 F 2 , F 3 given by (7) 



* "Teoria dolla elasticita," Cap. VI. 11 Nnovo Cimento, 1872; Annali 

 di Mat. (6) 1875. 



