168 Dr. L. Silbersteiu on Radiation 



will find that all the negative as well as the first and second 

 positive, and all the even powers o£ it, cancel mutually, 

 thus giving the series 



G (u) = a z u* -f a 5 u 5 + (10) 



which turns out to be very convenient for fractional values 

 of u, and even up to u=l'2. The coefficients are 



-W 1 _!_ X \ 16 X / - 2 M 



and so on. For the first five of these I find [a 3 = 2/3, 

 a 5 =— 7/75, a 7 =74/(3.5.7) 2 , &c], in decimal logarithms, 

 to five figures, which will be found convenient for purposes 

 of numerical calculation, 



log a 3 = 1-82391, log (-a 5 ) = 2*97004, 



log a 7 = 3-82685, log(-a 9 ) = 4*47401, loga„ = 6-95402 



} (11) 



The calculation of the following coefficients is left to the 

 reader. What is still required is the value of A, to be 

 substituted in (8). This coefficient, which is easily found by 

 means of (6 a), is given, in general, by 



A = - a * eQ 2^W) C ° S * W V= Mg(u) - 1 ' (12 > 

 where g(u) is as in (7), ±h(u)=u 2 sin u + (K— l)ng(u), and 



u na Ima 



The expression for li(u) can be introduced into that of tan 77, 

 and so on, and the resulting expression for A is easily con- 

 densed into a more elegant shape, for any permittivity and 

 for any wave-length X. But in the present paper we shall 

 be concerned only with two particular cases of (12), 

 viz. : 1st, the case of moderate permittivities, which is, 

 essentially, that of K = 1 ; and 2nd, the case of very 

 large permittivities, such as are required for line-spectra. 

 These two cases will be treated separately and, for the 

 present, as concisely as possible, in the following two sec- 

 tions. With regard to physical applications, it will be well 

 to remark here, in passing, that in the former case the 

 " source " and its functions would represent the cooperation 



