Source on the Aiiis of a Cylindrical lube. 661 



Tiie equations (27) will be satisfied by 



v == - <- , w== — ^— , w = — -x , . . (.SJJ 

 o e o// a - 



provided c 2 s {£/■* ,. n , 



0= -. ; == — f . r . ..... lOL) 



r ike -\- fjb lr 



Thus <f> must satisfy the equation (\/ 2 -\-h 2 )cf> = 0, or, with 

 the same coordinates as before, 



|!* + I|* + |!* +/A6 =o. . , . (32) 



Assurning $ to vary as e~~ m * g 9 the typical solution of (32 ) is 



cf> = AJ (k s r)e- ms * e l/i0 \ ..... (33) 



where k s 2 = mf + h*. ....... , (34) 



The condition of zero normal velocity over the surface of 

 the cylinder gives as before 



j Q '(k s a)=0 T . .. , . , , (35) 



Thus the solution of (32) which is appropriate to a positive 

 progressive wave, and gives tiie correct value for the flux 

 across the section z = 0, is 



^"ow write m s = a + l/3 s , ili = sc +if3, , '. . , (3 J) 

 whence 

 **-P*=k*-i», 2*J3,=kiifc, a *-/3>=-P, 2«0=i / */«, 



.. . .. , (38) 



Substituting from (37) in (56) we get for the value of *£ 

 In real terms 



_ g~ ag {acos (kct-j3z) + sin (kct-^fiz)} 

 *"~~ 27ra*(a* + £«) 



^ g- CT ^{« g cos ( fo* -fe) + ft sin (*c* - & :)U (k?' ) .« Q . 

 ' %waX** + p*){Jlk s a)¥ -'W 



The values of u s and a found from (38) must of course 

 be taken with the positive sign, to insure fmiteness at infinity. 

 It is easily seen that in the case of no friction when /x = 0, 

 the above result reduces to that previously obtained, 

 Phil. Mac,. S. 6. Vol. 24. ^ T o. 1.42. Oct. 1912. 2 X 



