108 PEOF. A. E. H. LOVE ON THE TRANSMISSION OF 



proportional to the resistance and to the inverse square of the wave-length. For 

 higher resistances and shorter waves the specific inductive capacity of the conductor 

 affects the value of A sensibly. For sea- water under air, and a wave-length of 5 km., 

 the value of I/A, the distance in which the amplitude of the waves is diminished in 

 the ratio 1 :e, is 478 x 10 6 , lengths being measured in kilometres. From the discussion 

 of this limiting case of the general problem it has been inferred that increased resist- 

 ance would be unfavourable to long-distance transmission, while increased wave-length 

 would be very favourable. 



4. Returning now to the general problem stated in 2, we shall take the axis of 

 the doublet to be the axis of z. The system being symmetrical about this axis, it is 

 appropriate to use the function II introduced by HERTZ,* and sometimes called the 

 Hertzian function. 



Let p, z, (f> be cylindrical co-ordinates, the senses of increase of z and <}> being those 

 of translation and rotation in a right-handed screw, and p denoting distance from the 

 axis- of z. Let E p , E_,, E0 denote the components of electric force, measured in 

 electrostatic imits, and estimated in the directions of increase of p, z, $ ; and let 

 H p , H;, H.J, denote the components of magnetic force, measured in electromagnetic 

 units, and estimated in the same directions. From the symmetry we have the 

 equations' 



E, = H, = H, = ........... (1) 



It will be sufficient to consider the case in which botli media are of magnetic 

 capacity unity, the dielectric is of specific inductive capacity unity, and the specific 

 inductive capacity of the conductor is neglected. Then one of the electromagnetic 

 equations is, in both media, 



_ _ 



c & cz " s p ' 



where C is the velocity of light, 3xl0 10 cm. per second. The remaining equations 

 are, 



in the dielectric, 



p ~u " : ' TT 3. 



U ct oz U vt p vp 



in the conductor, 



4 CE P =-^, 4CE,=1^( /0 H,); ...... (4) 



CZ p Cp 



where a- is the specific conductivity, measured in electromagnetic units. 



Now we are to suppose that E p , E,, H^,, in so far as they depend upon t, are 

 proportional to simple harmonic functions of period 2-7T/&C say, where 2Tr/k is the 

 wave-length, and we may take them to be proportional to e' kct . Then they can be 



* H. HERTZ, ' Electric Waves,' English ed., p. 140. 



