124 BELL SYSTEM TECHNICAL JOURNAL 



is the heat capacity per unit area or volume. Now by a general theorem 

 concerning spectral density matrices (see Appendix I) we have 



Jo 



giving finally by combination with (3.12) and (3.13), 



and 



Gkk' -, — 7/ ^kk' , (3.15) 





The spectral density 5(w) of R\{i) then becomes 



(3.17) 



If we are concerned with frequencies greater than a characteristic fre- 

 quency 



coo = AttW/L" (3.18) 



where L is the smallest of Li , t = 1, • • • , i', then the summation in (3.17) 

 may be replaced by an integration giving 



^( ^ _ ..+1 .-1 XD f \f(k)\'kUk 



Sic.) -2 rr ^ j -^^r^TDn^ ^^'^^^ 



where 



f^^^ = rr^^ [ f(r)e-"'-'' dr. (3.20) 



The integration in Eq. (3.19) is carried out over the entire iz-dimensiona? 

 /j-space. If the range of the function /(r) is sufficiently small compared with 

 the region ^, , or if we let A^ become indefinitely large, then the integration 

 in Eq. (3.20) may be extended to all of v-dimensional r-space. 



It is perhaps revealing to rephrase Eqs. (3.17) and (3.19) in terms of 

 distributions of relaxation times. In the theory of dielectrics we speak of 

 the real part of the dielectric constant being equal to a series of terms 

 summed over a distribution of relaxation times: 2,fltTi/(l + riw'), if the 



distribution is discrete, or / a(T)rJr/(l + tW), if the distribution is con- 



tinuous. In the above, o, is the weight for the relaxation time tj , and, in 



