with the Flow of Electricity in a Plane. 375 



tion of the given line. Similarly any line may be made an 

 equipotential line, either by making it bound a region of infi- 

 nite conductivity, or else by adjusting the potentials and posi- 

 tions of these same regions so that no part of the flow shall take 

 place along the given line. 



Such a sheet as we have imagined can be approximated to 

 by a piece of tinfoil cut to any shape, with any number of holes 

 in it, and having on its surface blocks of clean copper which 

 are connected with a thermo-pile or other means of maintaining 

 constant difference of potential. 



§ 2. The conditions of flow in such a sheet, however, are in 

 general unknown. But if infinitely resisting regions are absent, 

 and if the infinitely conducting regions are reduced to mere 

 points of infinite potential, we get the case of an unlimited 

 sheet containing point poles ; and in this all the conditions of 

 the flow are completely known for every point of the sheet. 

 Thus the potential at any point distant r l7 r 2 . . . from the poles 

 which emit quantities of electricity q l7 g 2 . . . in unit time, and 

 which tend to produce potentials </>!, </> 2 . . . at unit distance 

 from themselves, is (§ 33 of the former paper) 



Y =^-2is 2 ? 10 ^ < A > 



where 8 is the thickness and k the specific conductivity of the 

 sheet ; and from this expression stream-line equations and 

 resistance expressions follow. 



So, then, the problem of finding the flow conditions in any 

 bounded sheet containing point poles will be solved if we can 

 imitate the electrical boundary conditions in an unlimited sheet 

 by introducing extra point poles into it. These additional poles 

 are called images in the boundary, because they produce the 

 same effect in the given portion of the sheet as the boundaries 

 themselves produced — -just as the illumination inside a mirror- 

 walled room containing candles would be imitated in unlimited 

 space by placing extra candles at all the points occupied by 

 the images of the original candles in the mirrors. 



Images in rectilinear boundaries. 

 § 3. Now let it be required to cause a given straight line in 

 an infinite sheet of tinfoil to be a stream-line. All that is ne- 

 cessary is to arrange all the blocks of copper in the sheet sym- 

 metrically with respect to this line ; that is to say, they must 

 either be placed symmetrically upon the line, or they must 

 be placed in pairs one on each side of it and at equal distances 

 from it. It is usual to express this fact in the inverse way 

 vthus : — The image of a pole in a straight flow-line is a pole of 



