194 



It is important to note that equation (1) may also be deduced 

 from the general equation of hydrodynamics without its being 

 necessary to neglect the second power of the velocities, as is the 

 case in many problems of that kind. For an infinitely long time 

 of vibration i. e. for uniform rotation (1) simplities to 



d^oi 3 dot 



= \ (2) 



* (/r* r dr 



/» 

 The solution of (2) is «> = - -+• c„ c, and f, being integration- 

 's 

 constants. If the solid cylinder (radius R) rotates with uniform speed 



R'-^ . . . , 



i> in an iniinite liquid, the result will be cu = — —, giving lor the 

 ' r 



frictional couple as is well known the ex|)ression 



-AniiR'H (2') 



In order to arrive at a possible solution of (1) we have to make 

 our assumption regarding the motion of the liquid a little more 

 detinite by assuming that the angular displacement of each shell is 

 represented by 



«;•=/(') COS (pt — Cf (r)) (3) 



We may also consider (3) as the real part of the complex function 



ue'f', where ?/- is a function of r the module of which gives the 



amplitude of the oscillation and the aigument the phase-shift (({r). 



da 

 Remembering that ^ = v- equation (1) may be reduced to 



d^u 3 du ig pu 



1 A^. — (4) 



c/r* r dr ft 



Equation (4) is closely related to the differential equation of the 



cylindrical functions. Indeed by the substitution y = zv Bessei/s 



d'y \dy f l^ ,, , 



equation of the 1^' order --^ + ^ ,' + ( 1 - 7J // = ^' changes to 



d^v ^ dv 

 dz' z dz 



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



u=X\AJ,{ci)-\-BN,{ct)\, (5) 



r 



where c= 1/ - — ^, yl and B being complex integration-constants. 



