[ 37 ] 



IV. On some Problems connected with the Flow of Electricity 

 in a Plane. By Oliver J. Lodge, B.Sc. 



[Continued from vol. i. p. 389.] 



General form of the resistance-expression for two poles. 



§ 22. TT is convenient at this point to notice the general 

 J- form of the resistance expression for two poles A 

 and B in any bounded plate. It may always be written in the 

 form 



£*(?■_«) « 



where Q is a numerical quantity which depends on the general 

 form of the boundaries of the plate, and on the position of the 

 poles with respect to those boundaries ; while c is a number 

 which depends only on that part of the boundary which is infi- 

 nitely near a pole. As long as the poles are not placed on the 

 edge of the plate, c = l and is unaffected by any change in the 

 boundary ; but if either pole is placed on the edge, the angle 

 on which it is placed determines c, — the general law being 



77" 77" 



that if A is on an angle — , and B on an angle — , then c=m + n, 



(When both poles are inside the plate, both angles equal 2tt, 

 and therefore c=l, as stated above.) 



It is unnecessary to prove this rule formally ; but it is a con- 

 sequence of the necessary condition that A and B must be of 

 equal strength in the limited sheet ; for this condition requires 



2tt 2tt 



that if they are on angles ~— and ^— respectively, their 



strengths in the unlimited sheet must be in the ratio of 2m : 2n, 



AB 



and accordingly the quantity in (a) § 8 will occur to the 



2(m + n)th power. 



Hence, in the process of finding the resistance of any plane 

 conductor containing two small electrodes, the determination of 

 Q is the only difficulty. We have found it for the general 

 polygon of two sides ; let us proceed to find it for a few tri- 

 angles. 



Resistance of an isosceles right-angled triangle. Poles on the 

 equal angles. 



§ 23. This case was in reality the first figure with rectilinear 

 boundaries which I attempted. The positions of the images 

 of A are shown in fig. 5 ; they are on the corners of squares 

 covering all the plane. The images of B will be similarly 

 arranged, every source being surrounded by four sinks. 



