292 Methods for the Solution of Special Problems [OH. vm 



82. An infinitely long elliptic cylinder of inductive capacity K, given by = a where 

 .2?+i/y=ccosh(-f irj\ is in a uniform field P parallel to the major axis of any section. 

 Shew that the potential at any point inside the cylinder is 



1+cotha 



-Px 



A+COtha* 



83. Two insulated uncharged circular cylinders outside each other, given by 77 = a and 

 rj= -/3 where #+%r = ctan (-H'7), are placed in a uniform field of force of potential Fx. 

 Shew that the potential due to the distribution on the cylinders is 



, > Nm e^~ a) sinhMfl + fl-'fr+fl sinhrca . 

 2Fc > ( - ) n - r-f sin fit. 



V 



84. Two circular cylinders outside each other, given by 17 = and 77= -/3 where 



x + iyc tan J ( + 1*17), 



are put to earth under the influence of a line-charge E on the line #=0, y = 0. Shew that 

 the potential of the induced charge outside the cylinders is 



(a i?) 

 - L cos ^+ constant, 



the summation being taken for all odd positive integral values of n. 



85. The cross-sections of two infinitely long metallic cylinders are the curves 



where b>a>c. If they are kept at potentials Vi and F 2 respectively, the intervening 

 space being filled with air, prove that the surface densities per unit length of the 

 electricity on the opposed surfaces are 



, , ~ , j and 7 



4rra 2 log- 47r6 2 log- 



respectively. 



86. What problems are solved by the transformation 



d 



a- t 

 where a>l ? 



87. What problem in Electrostatics is solved by the transformation 



x+iy = cn(([> + fy\ 

 where -v/^ is taken as the potential function, <f> being the function conjugate to it ? 



88. One half of a hyperbolic cylinder is given by rj= q lt where 1 171 |<s-, and , rj are 



given in terms of the Cartesian coordinates #, y of a principal section by the trans- 

 formation 



x + iy = c cosh ( + irj). 



The half-cylinder is uninsulated and under the influence of a charge of density E per unit 

 length placed along the line of internal foci. Prove that the surface density at any point 

 of the cylinder is 



cosh ^ Vcosh2|-cos2j7!. 



