322 Steady Currents in Linear Conductors [OH. ix 



K and R be the capacity and resistance of the cable per unit length, these 

 latter quantities being supposed independent of x. 



The section of the cable between points A and B at distances x and 

 x -f dx is a condenser of capacity Kdx, and is at the same time a conductor 

 of resistance Rdx. The potential of the condenser is V, so that its charge is 

 VKdx. The fall of potential in the conductor is 



so that by Ohm's Law, 



dx iRdx (301). 



The current enters the section AB at a rate i units per unit time, and 

 leaves at a rate of i ' + - dx units per unit time. Hence the charge in this 



section decreases at a rate ~- dx per unit time, so that we must have 



_ / YKdx) = dx . ...(302). 



Eliminating i from equations (301) and (302), we obtain 



(303). 



368. This equation, being a partial differential equation of the second 

 order, must have two arbitrary functions in its complete solution. We shall 

 shew, however, that there is a particular solution in which V is a function of 

 the single variable xj\/t, and this solution will be found to give us all the 

 information we require. 



Let us introduce the new variable u, given by u = x Jt, and let us assume 

 provisionally that there is a solution V of equation (303) which is a function 

 of u onl. For this solution we must have 



t du*' 



dt du dt 2 \^ 3 du ' 

 so that equation (303) becomes 



du* = 



dV 



^- (304). 



ctu 



The fact that this equation involves F and u only, shews that there is an 



