i87s] STREAM LINES 57 



C/ i i \ sin a - ft _ v sin 20 



Sin I -i-^ A &amp;lt; T~-r~n ) T 



2 VPS PBV PC 2 r 2 



which since 



becomes 



sin 2 p - sin 2 a + sin a - j3 . sin a + /3 = 0, 



which satisfies (19). 



The chief interest lies in the cases where the cubic 

 breaks up into a straight line and a conic. This takes 

 place for one stream line of the system when all the 

 sources lie on a straight line, or when they form an 

 isosceles triangle with points of the same sign at the base. 

 The cases are 



1. Two Sources and a Sink. The conic is always a 

 circle with the sink as centre. If the sink lies in the line 

 of the sources produced, the radius of the circle is a mean 

 proportional to the distances of the sink from the sources. 

 If the sink lie between the sources, the circle is impossible. 

 If the sink is the vertex of an isosceles triangle, the circle 

 passes through both sources, and all asymptotes meet in 

 the point of zero flow furthest from the sources. If the 

 sink is half way between the sources, there are two straight 

 lines and a real and impossible circle. 



2. Three Sources of the same Sign. Every stre&quot;am line 

 has three asymptotes, meeting in the centre of gravity, 



and inclined at angles of -. If one of these asymptotes 



o 

 becomes a branch, the other branch is a hyperbola, with 



centre of gravity as centre, and axes in ratio of v/3 to i. 

 If the points form an isosceles triangle, the hyperbola 

 passes through the extremities of the base. If the triangle 

 is equilateral, the hyperbola coincides with its asymptotes. 



