KUNGL. SV. VET. AKADEMIENS HANDLINGAR. BAND 47. NO 4. 11 



e »ifc|U-M>and e-*l*K*-l*l>. 



Jtis physically evident that, whatever kind of frictional forces are imagined, their 

 ultimate action will consist in annihilating the disturbance due to the ridge for points 

 very far away from it. This will be effected, for a term: 



e i\k\().-\x\) } 



by replacing 



\k\ by \k\-~iu, 



!< being essentially positive, and for terms: 



e -t'|Å|(/.-|.r|) ; 



by replacing 



| Ar | by | k | -j- i/i . 



Passing from 1 to l bls by replacing in the terms of the second kind k by — k, we inf er 

 that, in order to have the physical meaning of P 1S for a 'positive x: 



The initially real and negative roots of D should have a small positive imaginary 

 part, and the real and positive ones should have a negative imaginary part. 



This is readily effected by transforming in I bls k in to: 



k {l — in). 



Again, for a negative x, you should pose: 



x = — | x | , 

 t hen in I you have terms: 



and on transforming to I bis , the same conclusion will be arrived at, viz.: 



The formulce I hls will be physically true if k is throughout imagined to be the limit for 

 u = o of 



k (l — ///). 



A positive real root of D should then always be imagined assituatedonthe»southern», 

 and a negative one on the »northern side » of the real axis of k. 1 

 The equation: 



Z> = o 1 (ocothfcÄcoth kh' + Q') — ij 2 gkQ{coth kh + coth kh') + g*k*(Q — q') =0 



has four real roots, two and two equal and of opposite signs. Denoting them by: 



± K and ± z , 



we may proceed to the calculation of the integrals by means of Cauchy's residue theorem. 

 Certainly there are also imaginary roots of D; the corresponding residues will, however, 

 necessarily decrease rather rapidly with increasing values of x, and may therefore be 



1 All this of course essentially iniplies that property of /(/) that it be zero for great values of )., or, more pre- 

 cieely, that it should tend towards zero more rapidly thau e - '"'^. 



