195 



t/j is the cjliiidrioal function of tlie I'*' kind and 1**' order, ^V, that 

 of the 2"^^ kind and 1»^. order '). 



As regards c an agreement must l)e come to. We shall choose the 



in 



root with the negative imaginary j)art i.e. c = /<? *, where ^'^ |c| = 

 As a first bonndarj'-condition we have Lim 7U(,. = 0''). As this 



;■ = CO 



relation must hold for all values of /, it follows that Iwi rn = 0, 



The cylindrical functions with complex argument all become infinite 

 at infinitj^ with the exception of the so-called functions of the 3"M<ind 

 or Hankei/s functions //^,^' and //y,'^'-^\ Of these Hp> 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 Hj^~'. By our choice of c in the negative imaginary half we are 

 led to the function ///'). For the integration-constants in equation 

 (5) this gives the relation B =z — iA ^), so that (5) becomes 



A 



u= ~ ///2) (cr) (6) 



r 



For the determination of A we have to use the 2"^^ boundary- 

 condition ait^=z a cos pt, R being the radius of the cylinder. We 

 therefore assume that there is no slipping along the wall. 



aE 

 Hence A = — ; , 



80 tiiat 



The symbol R is intended to indicate, that the real part has to be 

 taken of the function which stands after it. 



If we had chosen for c the root with the positive imaginary part, 

 we should have had to utilize the function ///^X It is quite easy to 

 verify that this would not have made any essential change in 

 the solution (7). 



1) Comp. Jahnke u. Emde. Funktionentafeln pp. 90 and 93. 



Nielsen. Cylinderfunklionen. Instead of iV Nielsen uses the symbol Y. 

 ') Prof. Verschaffelt puts Lim x,- = 0^ which in my opinion is not quite correct, 



?■= 00 



as the linear velocity lias to disappear at an infinite distance. Comm. li'Sb p. 22. 

 ') Between J, N, and H a linear relation holds. Gomp. J. u E. p. 95. 



13* 



