1873] STREAM LINES 49 



can be reduced. For if h, k be the co-ordinates of P, the 

 equation becomes 



//y _ h\ 



If ( y -- ) denote the sum of all the combinations of 



\h - x/ m 



At . r&amp;gt; 



expressions - j-, taken m at a time, we may write this 



!, x n 



-&c. = (8), 



an equation of the n th degree if there be in all n 

 sources. 



The degree of the equipotential lines is also = n if there 

 be an equal number of sources and sinks. In general, 

 if there be m sources of one sign, and n - m of another, 

 and m&amp;gt;n-m, 2m is the degree of the equipotential 

 lines. This is one of many features which make it more 

 convenient to work with stream lines. 



It is obvious from equation (8), that every stream line 

 must pass through all the sources. Thus, the circle in 

 case (c), which passes through no source, is not a complete 

 stream line, the other branch being the straight line APP , 

 which passes through all the sources. Distinct stream 

 lines can intersect only at a source, for at no other point 

 can 20 be indeterminate. Where two branches of the 

 same stream line intersect the velocity is necessarily zero, 

 changing sign in passing through the point. The physical 

 meaning of a branch is that two streams impinge, and are 

 thrown off with an abrupt change of direction. 



The same result is easily found from the analytical 



condition for a singular point -.:- - -=- =0. 



ax dy 



For --=- = -=- = velocity parallel to axis of y, 



du dv , ., 



T- = -7- = velocity parallel to axis of x, 



or directly by differentiation 



