226 



Mr Sharpe, Liquid Motion from a Single Source 



this equation from (8) so as to get rid of a 1 , after dividing out by 

 (0 — ir) we shall get the following result, 



O^c-^ + S 



Clvn.i 



cos mir 



(9), 



which gives us the value of OB. Comparing (9) with (1) we see 

 that at the point B u x = as we should expect, so that ABA' cuts 

 xO produced orthogonally. If E be the point where the boundary 

 BA cuts the axis of y we see from (6) and (8) that OE is deter- 

 mined by the following equation, 



cr + S 



ay*' 1 

 m' 



_a m (m+ 1) 



sin (m + 1) 



fiair 

 ~2~ 



(10). 



As we shall always suppose c^ < 7r/2 and as the points B and E 

 must be within the circle r — a the equations (8), (9) and (10) are 

 very useful to determine the relations and limits of the constants 

 employed. It is true it is conceivable that the boundary BAD 

 might cut the circle in more than one point. Such cases could be 

 treated by the present method, but probably they would be very 

 complicated. We shall therefore exclude them at any rate at first, 

 and suppose the constants so chosen that equation (8) in a x shall 

 have one and only one solution. 



4. It will presently (Arts. 5 &c.) be proved that there are an 

 infinite number of cases where the least value of m in (5) is 

 unity. For a moment assuming this, we see that at points at a 

 great distance from the terms depending on this unity value of 

 m are the most important. From (7) the form of the boundary 

 at such points can be inferred from the following equation, 



cr sin 9 + /xa0 + a x a6 = casin Cfj + fia^ + a 1 aa 1 (11). 



If c is finite, we have an asymptote parallel to Ox and the 

 shape generally resembles a canal closed at one end. If c = and 

 the right-hand side of (11) is finite, 6 = a 1 and there will be an 



oblique asymptote which is parallel to OA, and the shape may be 



