from an Electric Source. 175 



arid since g and Gr contain besides w purely numerical co- 

 efficients only, e is a function of \/a alone, independent of e , 

 of the size of the source, and even of the choice of time- and 

 length-units. (We shall see in the next section that this 

 property, with a slight modification, belongs also to a source 

 of any permittivity.) 



For u long " waves, i. e. for small values of w, we have 

 by (7), 



9 (ic)=^w 2 - 5| io«+ - w*- ... 4 gtu 2 , 

 and, by (10), 



(l + io 2 )G(w) = w*[a z + (a 3 + a 5 )iv*+ ...] ==|w 3 , 

 so that (21) becomes 



~T"-©r (^ 



where N is a purely numerical coefficient. Thus with 

 increasing "long" waves e rapidly decreases to zero*. 

 Formula (21a) is valid up to terms of the fourth order, 

 since the next neglected term is in uy\ Up to terms of the 

 sixth order I find, from the above series, 



2tt 9 a . 33 



(l+g*> • • • . (**) 



which can be used up to ir = *90 or '95. Beyond this the 

 rigorous formula (21) must be employed. Since G(io) is, 

 by (9'), essentially positive, so is also e, as might have been 

 expected. It attains, between each pair of consecutive zeros 

 (w l = 1*19, w 2 === 7*73, &c), a maximum. 



* The reader will be familiar with formula) such as (24a) from the 

 current, " elementary," theory of the Hertzian radiator, which, by a 

 very rough treatment of the problem, gives, for the ratio of the energy 

 emitted per period to the original electrostatic euergy of the apparatus, 



_ 16tt 4 l 2 R'_ 

 € ~ 3 T 3 ' 



11' being the radius of each of the (larger) spherical conductors and I 

 the distance of their centres apart. See, for instance, Drude's ' Physik 

 des Aethers,' 2nd ed. (1912), p. 515. Thus, li'Z 2 takes the place of 

 our a 3 . For the actual data of Hertz's experiment (R'=15, /=100, 

 X=480cm.) Drude finds (per half-period '352, i. e.) e = *701, which is 

 certainly a huge prodigality. The reader will notice that X is here not 

 very lt long,' ; being, in fact, only about 9 times (l 2 ii') ls , and 4*8 times I, 

 i. e. less than five times the length of the axis of the " dipole " which has 

 conceptually to correspond to the actual oscillator. 



