384 Mr. 0. J. Lodge on some Problems connected 



other point, distant r 2 from the centre of the circle and 

 \/ {r\ — 27\r 2 cos (j> + rl) from the nearest of the in points, will 

 be equal to 



N /(n m -27« l cos m<f> + rT) (C) 



To apply this theorem to the present case we must consider 

 the images of A in" two sets: — one set including A itself and all 

 its images whose angular distance from it is some multiple of 

 20; the other set containingall the remaining images, which will 

 alternate with those of the first set, and will like them divide 

 the circle into n equal parts. The product of all the distances 

 of the first set from the point B will be given by (C) if we put 



77* 



m = n— 7j and <£ = a — /3: the product for the second set will 



only differ in having <£ = a +/3. From the symmetry of the 

 expression, it follows that the product of the distances of the 

 images of B from the point A will be the same as that of the 

 images of A from the point B. Hence the numerator of (a) 

 is determined. As for the denominator, the product of the 

 distances of the second set of images of A from the point A is 



2r" sin nu. 

 The product of the first set appears to be zero, since A itself is 

 one of the set ; but remembering that the point to which dis- 

 tances are properly measured is not the actual pole A but a 

 point on the circumference of its small electrode (§ 8), we see 

 that the product becomes 



neglecting high powers of p. So the whole product of the A's 

 from A is 2np?'\ n ~ l sinna; similarly the product of the B's 

 from B will be 



2npr^~ l sin n/3. 



Hence the resistance-expression (a) reduces to 



Qfin — 2» r » rCQS n ( a _ ffy + ^ Qfin _% T n ^ cog n ( a + £) + r *n) 



4?i 2 pVf " Vf - 1 sin nu . sin nfi 

 or, as it is often more convenient to write it, 

 1 



,(5) 



2ttk8 



log 



( r i I n ^2) n \ /r-i\ n To) n \ 



-| +- -gcosn,-^)(^ +^| -2cog»« + ftJ 



v. P 4-?>* sin wrj <air> oiR. 



4?r sin nu . sin n/3 



