276 Methods for the Solution of Special Problems [CH. vm 



two edges of the slit are taken and turned so that the angle 2?r, which they 

 originally enclosed in the TF-plane, is increased to 4-Tr, after which the edges 

 are again joined together. 



The upper sheet of the Riemann's surface so formed will now represent 

 the upper half of the Tf-plane, while the lower sheet will represent the lower 

 half. Two points P lt P 2 , which represent equal and opposite values of W, 

 say + TFo, will (by equation (255)) be represented by points at which z has the 

 same value ; they are accordingly the two points on the upper and lower 

 sheet respectively for which z has the value W*. 



A circular path pqrs surrounding in the TF-plane becomes a double 

 circle on the ^--surface, one circle being on the upper sheet and one on the 

 lower, and the path being continuous since it crosses from one sheet to the 

 other each time it meets the branch-line. 



A line aft in the upper half of the TF-plane becomes, as we have seen, a 

 parabola aft on the upper sheet of the ^-surface. Similarly a line a.' ft' in the 

 lower half of the TF-plane will become a parabola a' ft' on the lower sheet of 

 the ^--surface. The space outside the parabola aft on the upper sheet of the 

 ^-surface transforms into a space in the TF-plane bounded by the line aft and 

 the line at infinity. Consequently the transformation under consideration 

 gives the solution of the electrostatic problem, in which the field is bounded 

 only by a conducting parabola and the region at infinity. The same is not 

 true of the space inside the parabola aft, for this transforms into a space in 

 the TF-plane bounded by both the line aft and the axis AOB. It is now 

 clear that the transformation has no application to problems in which the 

 electrostatic field is the space inside a parabola. 



In general it will be seen that two points, which are close to one another 

 on one sheet of the ^-surface, but are on opposite sides of a branch-line, 

 will transform into two points which are not adjacent to one another in the 

 TT-plane, and which therefore correspond to different potentials. Conse- 

 quently we cannot solve a problem by a transformation which requires a 

 branch-line to be introduced into that part of the Riemann's surface which 

 represents the electrostatic field. 



Images on a Riemann's Surface. 



333. In the theory of electrical images, a system of imaginary charges is 

 placed in a region which does not form part of the actual electrostatic field. 

 When a two-dimensional problem is solved by a conjugate function trans- 

 formation, the electrostatic field must, as we have seen, be represented by 

 a region on a single sheet of the corresponding Riemann's surface, and this 

 region must not be broken by branch-lines. The same, however, is not true 

 of the part of the field in which the imaginary images are placed, for this 



