312-317] Conjugate Functions 261 



314. If ~- denote differentiation along the surface of a conductor, on 

 which the potential F is constant, we have 



dW 

 dz 



dU 



so that cr = - R . 



4-7T 47T OS 



The total charge on a strip of unit width between any two points P, Q of 

 the conductor is accordingly 



f 1 



J ad8 = 4^ 



315. If, on equating real and imaginary parts of any transformation of 

 the form 



U + iV = $ (as + iy) (234), 



it is found that the curve f(x, y) = Q corresponds to the constant value 

 V = C, then clearly the general value of V obtained from equation (234) 

 will be a solution of Laplace's equation subject to the condition of having 

 the constant value V=C over the boundary f(x, y) = 0. It will therefore 

 be the potential in an electrostatic field in which the curve f(x, 2/) = may 

 be taken to be a conductor raised to potential C. 



316. From a given transformation it is obviously always possible to 

 deduce the corresponding electrostatic field, but on being given the con- 

 ductors and potentials in the field, it is by no means always possible to 

 deduce the required transformation. We shall begin by the examination of 

 a few fields which are given by simple known transformations. 



SPECIAL TRANSFORMATIONS. 



317. Considering the transformation W = 2 n , we have 

 U+iV = (x+ iy) n = r n (cos nO + i sin n6\ 



so that V=r n sinn0. Thus any one of the surfaces r n sin n0 = constant, 

 may be supposed to be an equipotential, including as a special case 



r n sin n6 = 0, 

 in which the equipotential consists of two planes cutting at an angle . 



This transformation can be further discussed by assigning particular 

 values to n. 



