317, 318] Conjugate Functions 263 



n = . This gives x + iy ( U + iV ) 2 , so that 



x=U*-V\ 2/ 

 and on eliminating U we obtain 



Thus the equipotentials are confocal and coaxal parabolic cylinders, in- 

 cluding as a special case (F= 0) a semi-infinite plane bounded by the line 

 of foci. 



This transformation clearly gives the field in the immediate neighbour- 

 hood of a conducting sharp straight edge in any field of force (fig. 86). 



n = 1. This gives 



U+ iV= - (cos 6 - i sin (9), 

 and the equipotentials are 



rF=sin0 or # 2 + 2/ 2 -^=0. 



Thus the equipotentials are a series of circular cylinders, all touching 

 the plane y = Q along the axis x = 0, y = (fig. 87). 



II. TF=log*. 

 318. The transformation W=logz gives 



so that the equipotentials are the planes 6 = constant, a system of planes all 

 intersecting in the same line. As a special case, we may take 6 = and 

 = TT to be the conductors, and obtain the field when the two halves of a 

 plane are raised to different potentials. The lines of force, U= constant, are 

 circles (fig. 88). 



Fm. 88. 



If we take U to be the potential, the equipotentials are concentric 

 circular cylinders, and the field is seen to be simply that due to a uniform 

 line-charge, or uniformly electrified cylinder. 



