of Electricity in a uniform plane conducting Surface, 457 



equipotential circles, we have, for the resistance of a disk on the 

 edges of which the poles are situated, and also for that of the 

 space outside it, according to §§ 22 & 24, 



*=^J - R=^.lo S 5. . . (13) 



26. It may be observed that the method by which the 

 resistance was obtained above (§ 22) is essentially the same as 

 that adopted by Kirchhoff. The method by which we first ob- 

 tained the expression, however, was founded on the consideration 

 of the equality of the resistance of the curvilinear rectangles into 

 which the sheet is divided by the intersecting systems of lines of 

 flow and equipotential lines [vide Plate IX.). This process, which 

 may be stated as follows, we afterwards found to be somewhat 

 similar in principle to that employed by Smaasen. 



Let two circular electrodes be placed upon the sheet so as to 

 coincide with two of the natural equipotential circles, which for 

 simplicity we will assume to have the same radius (=/>). The 

 spaces between consecutive lines of flow are of equal resistance, 

 and may for the present purpose be regarded as so many inde- 

 pendent conductors connecting the two poles and combined in 

 multiple arc. Consequently the resistance of the sheet which 

 is made up of these spaces, is equal to the resistance of one of 

 them divided by their whole number. These spaces in their turn 

 are each composed of 2ra equi-resisting rectangles arranged in 

 series ; so the resistance of each space is 2n times the resistance 

 of one of the rectangles, 2n being (as is evident from Plate IX.) 

 the whole number of equipotential lines in the sheet which are 

 not obliterated by the electrodes. 



Now consider the resistance of one of the four rectangles at 



the middle of the figure. Its length OD = /=a^ (§20) (for, 



7i=0 giving the straight equipotential line through (§ 19), 

 the next one will be given by rc = l). Its breadth b, which in 

 fig. 3 is denoted by C Q, will be 



L nr—a , a 



a cot — - — = a tan -. 



Calling 8 the thickness and tc the conductivity of the sheet 

 as before, the resistance of the rectangle will be approximately 



b/c8 kS tan \ol /a + 1 



The number of equipotential lines between the electrodes is 

 2n if the electrodes are of equal size and coincide with the nth 



