of a Cylinder in a Viscous Liquid. 589 



Equation (4) is closely related to the differential equation 

 of the cylindrical functions. Indeed by the substitution 

 y = zv, Bessel's equation of the first order 



dhi ldy / 1\ 



-changes to 



dz 2 z dz 

 It follows that the general solution of equation (4) is 



v^iiAJ^+BN^r)}, .... (5) 



where c= a / &-, A and B being complex constants of 



integration. J\ is the cylindrical function of the first kind 

 and first order, Nj that of the second kind and first order *. 



As regards c an agreement must be come to. We shall 

 choose the root with the negative imaginary part, i. e., 



c = ke 4 , where k = \c\= \\ / — i . 



As a first boundary-condition we have Limra r = 0. As 



this relation must hold for all values of t, it follows that 

 limrw = 0. 



r=oo 



The cylindrical functions with complex argument all be- 

 come infinite at infinity with the exception of the so-called 

 functions of the third kind, or Hankel's functions H^ and 

 Hp 2) . Of these H (1) disappears at infinity in the positive 

 imaginary half-plane and on the contrary becomes infinite in 

 the negative half, whereas the opposite is true for H^ 2) . By 

 our choice of c in the negative imaginary half we are led to 

 the function Hf } . For the constants of integration in 

 •equation (5) this gives the relation t B = — /A, so that (5) 

 becomes 



u=£Hf>(cr) (6) 



For the determination of A we have to use the second 



* Cf. Gray and Mathews, 'Bessel Functions.' Nielsen, Cylinder- 

 funktionen ; Jahnke und Emde, Funktiontafeln. Instead of N, Gray and 

 Mathews use the symbol Y. 



t Between J, N, and H a linear relation holds. Cf. Jahnke u. Emde, 

 p. 95. 



