Kotik and Lurye 

 Hij(cr) = H.jCa) - icrp^Jc/). n-Aj dS + pJJv„-V0in-Aj dS + /^ J JgiO-;!. dS 



(2.27) 



+ ^ f r[^i-3>^f -^(V^^)] n.;:.dS i = 4,5, 6 

 K i = 1, 2, 3 



H..(ct) = Hi .(cr) - io-p J J 0. rxn .;ij_ 3 dS 



+ P r fcV^ • Vc^i) Fxn •;!. _3dS + pj J g. rxn.;i. gdS 



^^r r[^i'^^^°'^] ^><"-^i-3ds i = 1.2,3 (2.28) 



s^-' j = 4, 5, 6 



H.j(CT) = Hij(o-) - icrpj jc/). fxn-^. _3dS 



+ P f fCVo • V^i) Fxn -/l. ,3dS + p r fg. rxn-Mj.gdS 



+ i&rr[^i-3x^~-^^V)] Fxn.Mj.3dS i = 4,5, 6 (2.29) 



sf j = 4, 5, 6. 



In all of these, the real and imaginary parts of H^j(cr) satisfy Eqs. (2.25) 

 and (2.26). 



We conclude with the following remarks: 



1. The Kramers-Kronig relations imply that any symmetry property in i 

 and j possessed by the element h?j is shared by h{j and vice-versa. Thus one 

 need only establish such a property for the real or imaginary part alone. 



2. It is known [4] that a submerged body oscillating in a stream can for 

 certain modes, frequency ranges, and speeds acquire energy from the stream 

 as a result of the oscillation (negative damping). The question then naturally 

 arises whether the Kramers-Kronig relations can still hold if over some part of 

 the frequency range negative damping occurs. Highly tentative considerations 

 indicate that there is at least a possibility of deriving a modified form of the 

 Kramers-Kronig relations in the case of negative damping; however, no firm 

 conclusions have been reached as yet. 



414 



