540 THE ELECTROMAGNETIC FIELD. [FT. III. CH. XIII. 



representation by taking the difference of ordinates of the outside 

 curve in Fig. 98 and the same curve pushed to the right the 

 distance r. The other curves in Fig. 98 represent the potential 

 at x when the battery is applied for times 1, 2, 3, 4, 5, 6, 7, times 

 KRx*. 



Since any derivative of a solution of (6) is a solution, the 

 derivative of (14) by x is also a solution, and 



F= 



represents the result of instantaneously connecting a battery and 

 then insulating the end of the cable. The distribution at any 

 time is of course shown in Fig. 97, and while V is initially infinite 

 at the origin, the total charge 



.00 



K \ 



Jo 



Vdx = q 



is finite, and remains constant throughout. 



255. General case of Telegraphic Equation. The 



telegraphic equation (5) has been treated byHeaviside, Poincare*, 

 Picard f, and Boussinesqj. We shall give the solution of Bous- 

 sinesq, not only because he has given the general solution of the 

 more general equation 250 (2), but because his method obtains 

 the solution by an ingenious artifice from Poisson's solution 247 

 (22), and the knowledge of other methods required by the processes 

 of Poincare and Picard is unnecessary. 



Let us put 



1 R 7 



so that our equation is 



* Poincare\ " Sur la propagation de I'electriciteY' Comptes Rendus, 117, p. 

 1027, 1893. 



t Picard. "Sur liquation aux derives partielles qui se rencontre dans la 

 th^orie de la propagation de I'61ectricit6. " Comptes Rendus, 118, p. 16, 1894. 



$ Boussinesq. "Integration de liquation du son pour un fluide indefini." 

 Comptes Rendus, 118, p. 162, 1894. 



