Theory of Faults in Cables. 73 



V 3 by multiplying them by rj, r 2 l, and si re pectively. Finally 

 add together all the results, right and left sides respectively, 

 excepting for V 3 , which must be subtracted, and then equate 

 the two sums. The result is 



*) J # 1 ** X 2 



+ pA^r'M;^ + r 3 A i M , ; , M. , :;da! + . . . + a^bnjn; 

 +A j ,r 2 ffl;'N;:-A i , s ffi;"E';}. (i8) 



It will be found, on making the substitutions in (18) of the 

 expressions for the M's and N's, and effecting the necessary 

 reductions, that in the summation on the right-hand side of 

 (18), the complete coefficient of every one of the A's vanishes 

 identically, by reason of equations (10) and (11), except for 

 A. ; whence 



f*. r* -V s iffl:" 



.... (19) 



This completes the solution; and the state of the whole 

 system is determined for any time t. 



15. When the initial potentials Vo^ &c. of the line in the 

 different sections are given explicitly as functions of x, the 

 sum of the integrals in the numerator of (19) may be written 



j" V to , sin (^f + fc) dx + f V, ^q\ sin Ei(£=f>) dx 





„ . aAx x 2 ) j 

 *2* 3 9i sm ; dx + . . . 



There is a great simplification when the initial state of the 

 system is, not arbitrary, but such as would be finally produced 

 by a constant electromotive force E acting at P (fig. 4). Then 

 the complete numerator of (19) reduces to 



E/cos&i 



