526 



XL HYDRODYNAMICS. 



The equipotential lines give a set of circles all tangent to the Y-axis 

 at the origin , while the lines of flow are a similar set all tangent 

 to the X-axis (Fig. 165). The water flows in on one side of the 



Fig. 165. 



origin and out at the other as if there were a source on one side 

 and an equal sink on the other close together. 



The function z n , of which the two examples just treated are 

 particular cases, gives an interesting case which is most simply worked 

 out by the introduction of polar coordinates. 



x = r cos G), y = r sin o>, 



z = x + iy = r (cos co + i sin CD) = re i(a , 



-f 



from which we obtain the two conjugate functions 

 126) u = 



If we multiply these two harmonic functions by constants and add, 

 the sum 



127) 



r n [A n cos (wo) + -# sin n& 



is the circular harmonic function treated in 140. We may accord- 

 ingly develop the velocity potential in a series of circular harmonics, 



