18 Dr. C. E. Weatherburn on Two Fundamental 



§ 3. A Particular Integral. — -An unlimited number of 

 particular integrals may be found to the equation (1). In 

 fact, if B is any biharmonic vector function, it is easily 

 verified that 



k 

 s = v 2 B— — — rgrad div B 



satisfies the equation. Taking ^ar as the biharmonic func- 

 tion, where a is any constant vector and r the " radius vector " 

 measured from a fixed point p, we obtain the particular 

 solution 



a k 



* = r~ 2(1+1) S raddiv ( ar )« • • • 0*) 



This is the vector equivalent of the set of integrals due to 

 Somigliana *. There is no loss of generality in taking a as 

 a unit vector. This integral becomes infinite at the pole p 

 to the same order as 1/r. Hence it cannot be regarded as 

 an actual solution of any physical problem for the elastic 

 body embracing the point p ; but we are able to construct 

 other integrals with s as a basis, which are finite and con- 

 tinuous throughout the body. 



It is important for our argument to notice that s (£><?), 

 which has been defined as a function of the variable point q 

 with p as its pole, is symmetrical in p and q. For if r is the 

 radius vector from p to q 



a k fa 



r 2(1 + k) 



a _ r a . rn 

 lr ?~r~] 



which is unaltered if r changes sign, that is if r is measured 

 from q to p. The expression is therefore symmetrical in 

 p and q, which may be expressed 



s (pq) = B (fi[p). 



In the potential theory -7- (- ) is an integral of Laplace's 



equation forming the basis of double-stratum potentials. 

 To find an analogous solution of (1) we shall determine 

 by means of (2) the surface traction corresponding to the 



* li Sulle equaz. dell' Elasticity," Annalidi Mat. (2) t. 16 (1888). 



