from an Electric Source. 177 



as 10 3 , especially when line-spectra are aimed afc, will be 

 explained later. Meanwhile, in order to fix the ideas, sup- 

 pose that e/v and therefore also ujw is or! the order 10 s , 

 and the radius of the sphere of the order 10 ~ 8 cm., i. e. 

 molecular or atomic. Then, for the whole extent of the 

 physically interesting spectrum (with the exception, for the 

 present, of the admirable X-ray domain recently discovered), 

 w will be negligible in the presence of 1, although u 

 will mount up to many units. Under these circumstances 

 formula (26) becomes 



€=47ne Go<y' (2ba) 



and (25), after some easy reductions, 



j _ ce 2 c os 2 7) 

 a ic' 2 



cos u K 



ic tan 7j = — - — _i- — - 



g{u) + u? 



(• (25a) 



^Ve shall denote by w 1} u 2 , &c, the successive roots of 

 g(u) = 0, i.e. of tanu = u. The values of the first three 

 of these roots have already been quoted (footnote, p. 173). 

 A formula, due to Euler, convenient for the calculation of 

 the following ones, is 



.2 3 13 . 146 _ 7 , 07 , 



where Xi = (2i+l) - *. Beginning from / = 20 or so we can 



77" 



safely write ?^ = (2i + l) ^ . 



For u=zu u ?/ 2 , &c. we have tan?7=:o, and therefore, 

 rigorously, J = 0, just as in the case of unit permittivity, 

 and, since G(V)>0 always, also e = 0. The corresponding 

 wave-lengths X will be denoted by i> lf y 2 i & c -j these waves 

 are strictly internal; not a trace of them escapes from the 

 source. 



Since iv 2 varies comparatively slowly, the successive 



* The tables of Jahuke and Emde {loc. cit. p. 3) contain sixteen of 

 these roots, to four decimal figures. Formula (27) will be found in 

 Eider's ' Introductio in anal, inf.' ii. p. 319 ; also in Lord liayleigh's 

 * Theory of Sound,' 2nd ed. vol. i. p. 334. 



Phil. Mag. S. 6. Vol. 30. No. 175. July 1915. N 



