342 Mr. 0. Heaviside on the 



ofE.M.F. will be 



e =(Z Q + Z 1 + lL ( p + lR 1 " + m a »)C, . . (119) 



if e is the total impressed force in the circuit, R/ and R 2 " 

 the wire and sheath functions of equations (55) and (56), 

 Part II., on the assumption m=0, and Z , Z x the terminal 

 functions, such that V/C = Z 1 at z=l, and = — Z at z = 0. It 

 does not matter how e Q is distributed so far as the magnetic 

 field and the current is concerned. Let it then be distributed 

 in such a way as to do away with the radial electric field, for 

 simplicity of reasoning. The simple- harmonic solution of 

 (119) is obviously to be got by expanding Z and Z x in the 

 form R + Lp, where R and L are functions of p 2 , and adding 

 them on to the Z(R' + Up) equivalent of l(LoP + Ri" + IV); as 

 in equation (66), Part II. 



Regarding the free subsidence, putting e =0 in (119) gives 

 us the determinantal equation of them's ; and as the normal 

 H functions are definitely known, the expansion of the mag- 

 netic field can be effected . The influence of the terminal 

 arrangements must not be forgotten in reckoning A. 



In coming, next, to the more general case of equation (56), 

 but without restriction to exactly longitudinal current in the 

 conductors, it is necessary to consider the transfer of energy 

 more fully. In the dielectric the longitudinal energy-current 

 is still YC. The rate of decrease of this quantity with z is to 

 be accounted for by increase of electric and magnetic energy 

 in the dielectric, and by the transfer of energy into the con- 

 ductors which bound it. Thus, 



dz dz dz 



But here, 



-g=SV, and-g=L C + E-F, . (120) 



by (59) and (56), Part II., E and F being the longitudinal 

 electric forces at the inner and outer boundaries of the dielec- 

 tric (when there is no impressed force). So 



-|-YC=SYV + L CC + EC-FC. . . (121) 

 dz 



The first term on the right side is the rate of increase of the 

 electric energy, the second term the rate of increase of the 

 magnetic energy in the dielectric, the third is the energy 

 entering the inner conductor per second, the fourth that 

 entering the outer conductor ; all per unit length. 



If the electric current in the conductors were exactly Ion- 



