Lord Rayleigh : Hydrodynamical Notes. 189 



The right-hand member o£ (3) may also be expressed in a 

 series of sines of the form 



2a) = Sa)/7r.2?2 -1 sin n7rd/u, . . . . (6) 



where n is an odd integer ; and thus for all values of n 

 we have 



_#R dR n « B 4 » 



?-■ 



dr 2 dr « z > i7r 



The general solution of (7) is 



E„= A„,'"/« + B.r— /- + ^VU-(-ip 



n n « '/I7r( 4^ — 7^71^) 



(8) 



the introduction of which into (4) gives ^. 



In (8) A n and B yj are arbitrary constants to be determined 

 by the other conditions of the problem. For example, we 

 might make R n , and therefore yjr, vanish when r = ri and 

 and when r=r 2 , so that the fixed boundary enclosing the 

 fluid would consist of two radii vectores and two circular 

 arcs. If the fluid extend to the origin, we must make 

 JB =0 ; and if the boundary be completed by the circular 

 arc r==l, we have A ?J =0 when n is even, and when n 

 is odd 



k+ ■ M 8 r\ ,x =o (9) 



Thus for the fluid enclosed in a circular sector of angle a 

 and radius unity 



mrla o zi 



I Q 2^ T ~ T ' nlT " /IAN 



^=8ft)a 2 S 1-^— 2 — T-sS sin ' " * ^ 10 ) 



1 nirynrir — 'ko?) a 



the summation extending to all odd integral values of n. 



The above formula (10) relates to the motion of uniformly 

 rotating fluid bounded by stationary radii vectores at = 0, 

 Q — ol. We may suppose the containing vessel to have been 

 rotating for a long time and that the fluid (under the 

 influence of a very small viscosity) has acquired this rotation 

 so that the whole revolves like a solid body. The motion 

 expressed by (10) is that which would ensue if the rotation 

 of the vessel were suddenly stopped. A related problem 

 was solved a long time since by Stokes *, who considered 

 the irvotational motion of fluid in a revolving sector. The 

 solution of Stokes's problem is derivable from (10) by mere 



* Camb. Phil. Trans, vol. viii. p. 533 (1847) ; Math, and Phys. Papers, 

 vol. i. p. 305. 



