The flow due to the source, vortex or dipole may be regarded as superposed upon a flow that 

 has no singularity at 2 = c. 



If a transformation is now made to the plane of a new variable C - f (2), then any line 

 source or vortex preserves its nature and strength on the ^-plane, provided it occurs at a con- 

 formal point for the transformation at which and near which dC,/dz exists and does not vanish. 

 A dipole also transforms into a dipole, but, in general, with a different moment and axial di- 

 rection. 



For, if y = /(c), so that y is the point on the ^-plane corresponding to s = c, 



B\x\(C - yj = Bin \f{z) - /fcj] = filn (z - c) + Bin '^^■' ~ '^^^ 



z - c 



The last term reduces to Bin [df/dz ] at s = c and hence represents a regular function at and 

 near this point; and the coefficient of In (^- y) is the same as that of In (2 - c). Thus a 

 source and vortex are conserved in the transformation. 

 Similarly, 



/i'e'*' , .„' z -c 1 

 = /i e 



in which 



C-y "^ f(3)-f(c) Z-C 



f(^) - f(c) ^df^^dC 

 Z-C dz dz 



as 2 -» c; 

 or, 



C-y Z-C 



where fi'= t\i and'*<'= (*+ 9, r and tj being modulus and amplitude of dC,/dz = re' . Thus a 

 dipole transforms into a dipole with its strength increased in the ratio of the modulus, and 

 its axial inclination to the real axis increased by the amplitude, of the transformation. 

 (For notation; see Section. 34) 



60. LINE SINGULARITY IN AN ANGLE 



Consider the transformation 



4:=2", 2 = C^/", [60a,b] 



134 



