276 Mr. W. G. Bickley on some Two-Dimensional 



§ 3. The method of sources used in the preceding paper is 

 not effective when the motion due to rotation of the bo undary 

 is desired. For that purpose a method outlined in a recent 

 note* by the present writer can be employed. The arc being, 

 as before, that part of a circle of unit radius given by 

 z= — t,e l( *, for which — a^#<a, the doubly connected space 

 outside is transformed into the up|.erhalf of the f-plane by 

 the transformation 





(5) 



where c = cos Ja. On the arc, we have^ = ±(o \ z — z | 2 , where 

 to is the angular velocity, and z the axis of rotation. The 

 choice of this axis is a matter of convenience, and is for 

 simplicity chosen as z = — l. So that on the f-axis of the 

 f-plane, on using (5), we get 



16s 2 £ 2 

 t=i*>v(f+l)<+4^ ^ 



The corresponding value of w, free from infinities in the 

 upper half of the z-plane, is then, except as to an irrelevant 

 constant, 



to 



"wj_. 2<B c«(p+i)»+4««p't-r * * ( ; 



The integral is easily evaluated by the method of residues, 

 and may be expressed in the three forms : — 



c£ 2 + 2i£—c } v ' 



....... («c) 



1 + se 



by the use of (5) and (1) above. Form (8 c) is convenient 

 to enable the stream-lines to be drawn, and these are given 

 for the case of a semicircle in fig. 5. An alternative, but 

 special, method of obtaining the result is furnished by the 

 fact that a rotation about the centre of the circle leaves the 

 liquid undisturbed. This may be regarded as instantaneously 



* Phil. Mag. June 1918. 



2z 



o -2r 1 + *e T 



