1873] STREAM LINES 55 



Remembering that 



d 2 u _ d 2 u _ / si 

 ^~ 2 ~~Sv2~ V ~ 



we can readily bring (17) into the form 



sin 20\^fcos (6 + tf)\ /cos 20\^(sm (0 



or, 



In this last expression and 8&quot; may assume the value 0. 



The radius of curvature may be similarly expressed, 

 but such expressions can hardly have a practical applica 

 tion. 



The cases of practical interest are mainly those where 

 the number of sources is small. We have already examined 

 the cases of two sources of the same or opposite signs. 

 We will now proceed to consider the cases that arise when 

 there are three or four sources. 



Three Sources. In general the curves will be cubic 

 passing through the three sources, and having asymptotes 

 determined as above. The direction of flow at any point 

 of the field may be found by observing that if &amp;lt;f&amp;gt; be the 

 angle between the tangent and a radius vector, 



It will sometimes be possible to find the direction of flow 

 geometrically by the following obvious theorem. 



If a circle be described touching a stream line at any 

 point, and cutting off from the radii vectores of that point, 

 fractions of their lengths, ^ // etc., where ^ is negative 

 if the point of intersection is in the radius vector produced, 

 and also negative if the radius vector is drawn from a sink, 

 then 



ZM-0. 



When the number of sources is large this theorem is 



