386 Mr. 0. J. Lodge on sorne Problems connected 

 If at the same time A is on OX or OY, this becomes 



•^MHT) (10) ' 



A direct application of Cotes's property and equation (B) 

 to this last case gives us as the perfectly accurate expression 



whatever be the size of the electrodes, provided that they be taken 

 of such shape as to fit the equipotential lines passing through 

 the two points P and Q to which distances are measured. Both 

 these points are on the line A B, one of them distant p x from 

 A, the other p 2 from B. This expression will reduce to (10/ 

 when pi and p 2 are regarded as small quantities of the first order. 

 § 14. Although all these expressions are proved only when 6 

 submultiple of tt, vet by continuity they would seem likely 



is a 



to hold true always. If so, we have expressed the resistance 

 of a Riemann's surface of any number of leaves, but all con- 

 nected at one branch-point 0. 



General resistance of a " Strip" or regular two-sided polygon 

 poles anywhere. Fig. 4. 



§ 15. The strip being a special case of the 

 wedge, its resistance will follow directly from 

 (5)* by making 6 vanish. We must substitute 



s — r^j a=;r x u } b = r 2 fi ) c — r^—r 2) 

 and then make r x =r 2 + c infinite, keeping c, a? 

 b y s all finite (fig. 4). The substitutions give 

 us 



s } s r 2 7 



so the resistance (5)' becomes 



dl= —log 



e 



2ttkS 

 (cosh 



it 



a—b V , c a + b V 



— COS 7T )( COSh - 7T— COS IT ) 



S A S 8 / 



ttY 



. air . bir\ 



sin — sin — 

 s s 



.(ii) 



* It might be objected that when n is made large, higher powers of p in 

 the expansion of (r—p) n may not be neglected, as was done in the process 

 of obtaining (5). This would be true if the size of n were merely due to 

 the shutting up of 6 with a constant r ; but in our present case r is also 

 large, and the expansion is in descending powers of r. 



