175, 176] STEADY FLOW IN CONDUCTORS. 347 



finite extent and bounded on one side by a plane perpendicular to 

 the cylinder, Fig. 70. We may obtain an upper and lower limit for 

 the desired correction by an artifice due to Lord Rayleigh *. It is 

 evident that if we introduce anywhere a portion of conductor of 

 greater conductivity we increase the conductance of the whole. Let 

 us accordingly introduce in the mouth of the cylinder a plane sheet 

 of infinite conductivity, thus rendering that circular section equi- 

 potential. The flow in this case will resemble the actual flow in 

 that V will be continuous in crossing the plane, while it will differ 

 from the actual case in that dV/dn will be discontinuous, its 

 integral over the section, or the total current being continuous. 

 We may then use for the portion below the mouth the solution for 

 a straight field, so that the resistance of a length I of radius a is 



i i 



Above the mouth of the cylinder we may use the formula for the 

 flow from a circular disk of radius a to infinity, so that the resist- 

 ance on the upper side is 



R- l 

 ~i5x' 



Consequently the lower limit of the resistance is 



In a similar manner the resistance of the system will be increased 

 if we introduce non-conducting surfaces not coincident with the 

 walls of current tubes. Let us below the mouth of the cylinder 

 suppose the cylinder split up into an infinite number of cylinders 

 of infinitesimal cross-section, by means of cylindrical non-conducting 

 surfaces introduced, and let the current density in these filaments 

 be maintained constant, in the whole of the cylinder. Then 

 below the mouth the equipotential surfaces will be planes, but on 

 the upper side of the plane of the mouth the potential will not be 

 constant, as we shall show. Consequently at the mouth of the 

 cylinder V is discontinuous, while d Vfdz is in this case continuous. 

 Since below the mouth 



= x dF 



dz 



* Rayleigh, Tlieory of Sound, Vol. i. 305. 



