Lord Rayleigh : Hydro dynamical Notes. 191 



A similar transformation is required when a = 37r/2. 

 When a = 7r. the boundary becomes a semicircle, and the 

 leading term (?i = l) is 



th = -^rsm0=-^ (14) 



which of itself represents an irrotational motion. 



When a = 27r, the two bounding radii vectores coincide 

 and the containing vessel becomes a circle with a single 

 partition wall at # = 0. In this case again the leading term 

 is irrotational, being 



f/co=-^ r ^smie (15) 



Steady Motion in a Corner of a Viscous Fluid. 



Here again we suppose the fluid to be incompressible and 

 to move in two dimensions free from external forces, or at 

 any rate from such as cannot be derived from a potential. 

 If in the same notation as before i/r represents the stream- 

 function, the general equation to be satisfied by ^ is 



V'f = 0; (1) 



with the conditions that when = and #=«, 



yfr=Q, dfjd6 = (2) 



It is worthy of remark that the problem is analytically the 

 same as that of a plane elastic plate clamped at 6 = and 

 6 = a, upon which (in the region considered) no external 

 forces act. 



The general problem thus presented is one of great diffi- 

 culty, and all that will be attempted here is the consideration 

 of one or two particular cases. We inquire what solutions 

 are possible such that -v/r, as a function of r (the radius 

 vector), is proportional to r m . Introducing this supposition 

 into (1), we get 



{ TO2+ £}{ ( " ! - 2)2+ £} t=0 ' • • (3) 



as the equation determining the dependence on 6. The most 

 general value of yjr consistent with our suppositions is thus 



^ = r m [Acosm0 + Bsmm6 + Cco${m-2)0 + Vsm(m-2)0}, 



.... (4) 

 whore A, B, C, D are constants. 



