I £ \°\1 



Problems in the Theory of Elasticity. 



21 



for the particular values i, j, k of a. The first term of (7) 

 substituted in (10) gives 



an expression which becomes equal to — ±7rT)(p)/(l + k) 

 when the radius of Z decreases indefinitely. The next term 

 of (7) contributes to the value of the integral 



~ i^k L D ( 5) ' L 1 X ( n X gTad r ) i+j X ( n X grad r ) J + • ' *] d$i 



which vanishes along with the radius of the sphere ; for since 



ii has the same direction as r it follows that n x grad- 



becomes infinite of order less than the second. Hence the 

 integral vanishes in the limit. Lastly, from the third term 

 we derive 



ifJz D «-fe{ i - (grad '; gradr)i }]it> s 



= if* j z D (') • (8>' ad ' S^d r) £ (i) ds. 



The limit of the value of this expression when Z decreases 

 indefinitely is —±wkD(j>)/(l + k). 



Substituting in (10) the value thus found for the integral 

 over Z we obtain the result 



■» 

 4ttD(p) = j D(j) . [F 2 (s P ) i+ F 2 (sp) j + F 3 (*p) k l ds 



[is l (ps)+jS2(ps) + ks s (ps)].Y{s)ds, . (11) 



■A 



which is the vector equivalent of Somigliana's formulse * for 

 the components of the displacement at any point of an elastic 

 isotropic body in terms of its surface values and the values 

 of the surface tractions. The formula (11) is analogous to 

 the relation 



giving the value of an harmonic function u at any point p of 

 * Annali di Mat. 17 (1889). 



