during the process of Sheathing. 177 



which the resistances are F„ F 2 , . . . F„, and their respective 

 distances (expressed in resistance also) from the same end of the 

 cable x v a? 2 , . . . x n , then we may calculate by formula (4) the 

 position of the resultant fault of the first two faults Fj and F 2 ; 

 from the first resultant fault and the third real fault F 3 , we may 

 calculate in the same way the second resultant fault, and so on. 

 At last we have calculated the position of the (n — l)th resultant 

 fault, which is in fact the resultant fault of all n faults. To have 

 an algebraical expression for this, we will call P the product of 

 the resistances of n faults, and F^ the resistance of a single fault 

 whose distance from the one end of the cable is x ; and calling 

 z the distance of the resultant fault of all n faults, expressed in 

 resistance and measurement from the same end of the cable, 

 we have 



— - x=x lX * (5) 





which may be easily calculated. 



Supposing now the insulation of a cable to be a certain 

 function of x the resistance of the conductor, which may be ex- 

 pressed by /(a?), then the resistance of a fault in each point of 

 the cable will be -et x \ 



¥ * = ~dx~'' 

 and thus we have, according to formula (5), 



(6) 



f ' dx 

 . Jo/I*)'* 



r. 



dx 



M 



I being the resistance of the conductor of the whole cable. 



If now a cable is insulated equally at all points, we have to 

 substitute in formula (6) 



f(x) = constant; 

 thus 



1 xdx 



._Jo = 



or the resultant fault of a cable with uniform insulation is in the 

 middle of the conductor resistance, which was evident a priori. 



Having now all necessary formulae, I may proceed to give the 

 proofs of the advantages named under 1, 2, and 3, which shall 

 follow in a subsequent article. 



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