of a Cylinder in a Viscous Liquid. 591 



For the reduction of the second part on the right-hand 

 side of (9) we make use of the well-known recurrence- 

 formula of the cylindrical functions : 



lO:) = 1 



dz ° z H i W- 



By its application (9) assumes the form : 



r3"*1 1.T " 2a »*_t H o 2) ( cR ) i P A nft x 



giving for the frictional couple : 

 K = 2^R3[|^] e = - 4^R^ + R ± [2^Wac |^^]- (U) 



For an infinite time of swing, i.e., p = 0, but with a 

 rotational velocity differing from 0, |c| = a/— becomes 0. 

 In that case the second term on the right of (11) disappears on 



*™H< 2> (cR) 



only the first term then remains, which agrees with (2 ; ). 



two grounds : first, because c=0, secondly, Lim (2) ^ =0 ; 



Moreover * 



T . H< 2 >(«R) . 



Lim — ,-tzz = — i. 



cTL=o Hf(cR) 



It appears from the accompanying graphs f of the modulus 



p Hf(cR) 1 \. ,. . . 

 and argument of — ^ that this limiting value is practi- 

 cally reached at 



jcRj=£.R=10, (12) 



where 



* Cf. Jahnke u. Emde, 1. c. 



t Tables for H^' and H^ 2) will be found Jahnke u. Emde, pp. 139, 140. 



