I dw I 



-7= — =M 

 I az I 



-1 



x^ -y'^ +2 ixy 



[(a;2-y2)^ +4 a;2 y2j 



[37d] 



j;'' +y^ 





Thus 7 = oc at the origin, where 3=0 and a singularity occurs. 



In terms of polar coordinates r and 6, with ^ measured from the a;-axis so that x = t cos 6, 



y = T sin d, 



"COS usin 6 



<?!, = ^^^ , i// = - ^^ [37g,h] 



r T 



Tlie equipotential curves and the streamlines are circles through the origin; their equa- 

 tions are obtained by assigning a constant value to or \h in Equations [37b, c] or in the equiv- 

 alent equations 



\x +y = — ^1^ -^[y^ — = — 5 [37i,]] 



^ 2^j 4c/,2' \ 24,) 4^2 



The radius is y./ {2 |(^|), or /i/(2|iy/|). See Figure 43, 



This is the flow due to a uniform line dipole or doublet. It is obtained in more elemen- 

 tary fashion in Section 15. The axis of the dipole is here the a;-axis, which represents a plane 

 of symmetry. The constant yi represents twice the point-dipole moment per unit length; it may 

 be called the line-dipole moment. The dipole axis is regarded as directed toward the side of 

 maximum ch\ if ^ > 0, this is here the pasitive a?-axis, if ^ < 0, it is the negative a!-ax is. 



The components of velocity in the directions of x and y, or of t and d, respectively, are 



a;2 -y2 2iixy cos sin d 



u = fi — J—, V = — —; q^^ y —p ■: lQ = \^ — — [37k,l,m,n] 



The conjugate flow represents a line dipole with axis along the y-axis, directed toward 

 negative y if ^ > C. It is also obtainable from the transformation 



") = --, cA = - ^ . i'/ = - ^ [37o,p,q] 



More generally, the transformation 



ye 



[37r] 



where y an 1 a are real and z = x +iy , represents a line dipole located on a line cutting the 

 a;y-plane at (x , y ), with its axis inclined at an angle a to the positive a;-axis; see Figure 44. 

 Since 



e'" cos a + ^ sin a (cos a + z sin a ) [x- XQ-i{y- y^)] 



76 



