that 



850 



, , b'R + 1 



L" — I .iN'b"ji I I 



1 



0) 



{b'R+iy+b'"K- 



(30) 



For a further approximation in the case, that the liquid may by 

 approximation be considered as unbounded, (24') can be developed 

 in the form of a series. For this purpose we write first: 



b-'R-'-SbR+d bR'+l _,^,^,_^ 



,, b'R' + 'dbR-rd b'R'-\-'óbR+'ó' bR^ 



L — ^-jiR''^,. ~. (24") 



bR^l bR-\ bR' + l ,„ „ ^ ' 



bR+1 bR'—l 



when e-2A(/?'-y?i jg sufficiently small"), formula (25) will hold as a 

 first approximation ; if necessary a first correction-term may be added 

 of the form 



' '{bR'+\){bR+\y ^ ' 



the value of which can be computed fairly easily, when an approximate 

 value has been found for 7j. 



1) If k (ki) is replaced by the conjugate imaginary quantity k.2 , it is c'ear, that 

 the real part ol' a and also of xr do not undergo any change (b^ and b.2 are 

 similarly conjugate), so that exactly the same results must be obtained, in particular 

 the same equations (30). That this is actually true may be easily seen from the 

 fact that Li and L^ according to ('24') are also conjugate imaginary. 



We might even, in general, have represented the damped harmonic oscillation 

 by the real part of 



a = «J 4" «2 = ajfi^'i' + a,e^'»'. 



We should then have obtained 



and have found, that a must satisfy the equation 

 d^'a da, da, 



de ^ ' dt ^ ' dt ^ 

 which, owing to L'.^ — L\ and L"o = — L'\ may also be written as: 

 rfV(' dd d(a", — «",) 



dt^ ■ dt ' dt 



By putting Oj = a^ a may then be real (form. (2) ). 



-) The coefficients of this factor in (24") canfiot become infinite in this case, 

 on the contrary they do not differ much from unity. 



