226 
Mr Sharpe , Liquid Motion from a Single Source 
this equation from (8) so as to get rid of oq, after dividing out by 
(6 — 7r) we shall get the following result, 
0 = c -^7 + ' s (^s- cos m7r ) ( 9 )> 
which gives us the value of OB. Comparing (9) with (1) we see 
that at the point B u x = 0 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, 
rrr 
cr + S — , -.x sin (m + 1) ^ 
_a m (m + 1) v 2 
pair 
~ 2 ~ 
( 10 ). 
As we shall always suppose otj < 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 cq 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 0 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 6 4- paO + a x ad = ca sin oq 4- pao q 4- a x aa x (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 = 0 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 
