358 Lord Rayleigh on the 



circumference. By (11) /3 = a, and 



(n + 2)A» + nB ll =0 (12) 



The normal velocity at the boundary is represented by 

 dty/dO, and we might be tempted, in our search after sim- 

 plicity, to suppose that this is sensible in the neighbourhood 

 of one point only, for example = 0. But in that case the 

 condition of incompressibility would require that the total 

 flow of fluid at the place in question should be zero. If the 

 total quantity of fluid entering the enclosure at # = is to be 

 finite, provision must be made for its escape elsewhere. This 

 might take the form of a sink at the centre of the circle ; but 

 it will come to much the same thing, and be more in harmony 

 with our equations as already laid down to suppose that the 

 escape takes place uniformly over the entire circumference. 

 This state of things will be represented analytically by 

 ascribing to ^ a sudden change of value from — 1 to +1 at 

 = 0, with a gradual passage from the one value to the other 

 as increases from to 2tt, or, as it may be more con- 

 veniently expressed for our present purpose, yjr is to be 

 regarded as an odd function of such that from = to 

 = 7r its value is 



ft-i-| (is) 



The symmetry with respect to = shows that we are 

 concerned in (10) only with the sines of multiples of 0, so 

 that having regard to (12) we may take as the form of yfr applic- 

 able in the present problem, 



f = ZC n slnn0{(n + 2)r n -?ir"+ 2 }, . . . (14) 



in which n is any integer and C n an arbitrary constant. It 

 remains to determine the coefficients C in accordance with 

 (13). Whenr=l, 



ylr = 2SC n smnd=l--; 



IT 



and this must hold good for all values of 6 from to tt. 

 Multiplying by sin m6 and integrating as usual, we find 



Cn = — ; (15) 



so that 



'7r.y]r = Xsmn0< (l+ r V ft — r"+ 2 | . . . 

 is the value of yfr expressed in series. 



