48 SUBMARINE CABLE LAYING AND REPAIRING. 



minute and therefore the resistance, as tested in the factory 

 at standard conditions, must not be less than 

 6,500 



18-25 



356 megohms per naut. 



The derivation of the necessary formulae for calculating the 

 relative dimensions and weights of copper and gutta-percha 

 may now be explained. 



The insulating coating on the conductor is, of course, 

 tubular in form, the inner surface being in contact with the 

 conductor and the outer surface with the metallic sheathing or 

 earth through the sea-water. Between these surfaces electro- 

 static induction takes place, and the leakage current (which is 

 a measure of the insulation resistance) passes ; both actions 

 taking place along directions radial to the tube. The resistance, 

 therefore, varies directly as the thickness of the insulating 

 material, and inversely as the mean effective area. The thick- 

 ness is equal to |^(D - d), and the mean effective area lies 

 between ttDZ and Trdl, where I represents the length of tube 

 and D and d the outer and inner diameters respectively. 

 Both the thickness and effective area are, therefore, functions of 

 the diameters, and it is easily shown that the ratio of thickness 

 to area is 



%^f^°«H <«> 



which is therefore proportional to the resistance. That is, 

 the resistance of a tube of any material (the lines of flow 

 being radial to the tube) is proportional to the Napierian 

 logarithm of the ratio of outer to inner diameters, and inversely 

 to the length multiplied by 27r. To obtain the resistance, 

 therefore, of any length of tube, we have to multiply the above 

 by the resistance of a piece of the material of unit thickness 

 and unit area — that is, the resistance between two opposed faces 

 of a cube. Thus, if I is expressed in feet the constant must 

 be the resistance of a cubic foot, and if in nauts the resistance 

 of a cubic naut. The cubic naut is most generally used, for 

 the simple reason that it is more convenient to put I in 

 nautical miles. It is not necessary to put D and d in any 

 definite units, because they are simply expressed as a ratio. 

 They must, however, be put in similar units. 



