KUNGL. SV. VET. AKADEMIENS HANDLINGAR. HAND 47. n:o 4. IT 



representing an upheaval which might be made steeper or flatterby changing t hc < -ons! ;uit «.. 

 The roots of J, apart from the real ones: 



±x, 



are all, as is easily shown from elementary properties of the function coth, purely imagi- 

 nary. Denoting them by: 



±kv=± il,. 



/,, will be the positive real roots of: 



a*g(cotl r ?i + cot l v h') + gl v {Q — q') = 0. 



In accordance with what is, for a symmetric ridge, an evident physical necessity, it 

 will be found, on applicating the residue theorem, that if 



0(1*1) 



be the result found for a positive x, that for a negative x will be: 



V(— |a;|) = — <b(\x\); 



so we need only write down the results for a positive x. 



Taking then, to begin with, the imaginary roots, the corresponding contribution 

 will be: 



00 



2 i v cl v Goe int C 2-2 7 ,i , r- 

 V vt v sin/, /t . J 1 (? /,,) / 



X 



X 



2i v< cl,age inl /' 2 . 2 , , ,. 

 Vrt* d vsml v h.J l (il v ) 1 



— x 



since the integration is to be conducted differently according to the sign of X — x. 

 Here j l (//,) is real and equal to: 





consequently, on taking the real part of the above expression, there will only be left the 

 terms containing 



sinat, 



which terms, as functions of x, will rapidly diminish with increasing values of that 

 variable; also evidently there will be no disturbance for x = o. Extending the results 

 to negative values of x, we may say: The imaginary roots will contribute a standing 

 oseillation with a node in the origin. 



K. Sv. Vet Akad. Handl Bd 47. N:o 4. 3 



