328 Steady Currents in Linear Conductors [OH. ix 



25. If n points are joined two and two by wires of equal resistance r, and two of 

 them are connected to the electrodes of a battery of electromotive force E and resistance R, 

 shew that the current in the wire joining the two points is 



2E 

 2r+nR' 



26. Six points A, B, C, D, P, Q are joined by nine conductors AS, AP, EC, BQ, PQ, 

 QC, PD, DC, AD. An electromotive force is inserted in the conductor AD, and a 

 galvanometer in PQ. Denoting the resistance of any conductor XY by r xy , shew that 

 if no current passes through the galvanometer, 



(r B c + r BQ +r CQ ) (TA^DF - r AP r DC ] +r BC (r BQ r DP - r AP r CQ ) = 0. 



27. A network is made by joining the five points 1, 2, 3, 4, 5 by conductors in every 

 possible way. Shew that the condition that conductors 23 and 14 are conjugate is 



(^15 + -#25 + 35 + ^45) (^12 ^34 - ^13 



where K n is conductivity of conductor rs. 



28. Two endless wires are each divided into mn equal parts by the successive 

 terminals of mn connecting wires, the resistance of each part being R. There is an 

 identically similar battery in every rath connecting wire, the total resistance of each 

 being the same, and the resistance of each of the other mn n connecting wires is h. 

 Prove that the current through a connecting wire which is the rth from the nearest 

 battery is 



$0(1- tan a) (tan 1 * a+ tan-' a)/(tan a - tan a), 

 where C is the current through each battery, and sin 2a = h/(h+R). 



29. A long line of telegraph wire AA 1 A 2 ... A n A n + i is supported by n equidistant 

 insulators at ^ A 2 ... A n . The end A is connected to one pole of a battery of electro- 

 motive force E and resistance B, and the other pole of this battery is put to earth, as 

 also the other end A n+1 of the wire. The resistance of each portion AA, A 1 A%, ... 

 A n A n + i is the same, R. In wet weather there is a leakage to earth at each insulator, 

 whose resistance may be taken equal to r. Shew that the current strength in A p A p + i is 



B cosh (2n+ 1) a + */Rr sinh (2w+2) a 

 where 2 sinh a = jR/r. 



30. A regular polygon AiA 2 ... A n is formed of n pieces of uniform wire, each of 

 resistance o-, and the centre is joined to each angular point by a straight piece of the 

 same wire. Shew that, if the point is maintained at zero potential, and the point AI 

 at potential F, the current that flows in the conductor A r A r + i is 



27 sinh a sinh (w-2r+l) a 



or cosh na 

 where a is given by the equation 



cosh2a = l+sin -. 

 n 



31. A resistance network is constructed of 2n rectangular meshes forming a truncated 

 cylinder of 2n faces, with two ends each in the form of a regular polygon of 2?i sides. 

 Each of these sides is of resistance r, and the other edges of resistance R. If the electrodes 

 be two opposite corners, then the resistance is 



tanh 6 



where sinh 2 = ^-5 . 



