170 Dr. L. Silberstein on Radiation 



The mean energy of the source, (13), can now be written 



U =yTF(w), (16) 



where 



2 | (17) 



It seemed interesting to consider the differential coefficient 

 of the mean energy with respect to the temperature, at 

 constant \,=\ c , say, 



= (BU/9T) V ..... (18) 



which might be called the sphere's " specific heat " for con- 



4 

 stant frequency. By (16), and writing -7 = Coo, we have 

 ultimately 



C/C„=?[F(«,)- W F'(«0], • • • (19) 



F being given by (17). In series form we can write, by 

 the developed form of (17), and remembering that a 3 = 2/3, 



C/C ao = l-b 2 w 2 + b i io i -b 6 iv 6 + (19 a) 



to = T;/T, (20) 



where T c stands for the constant 2iraj\ c . The coefficients b+ 

 which — as far as I have calculated them — are n\\ positive, are 



3 9 15 



&2 = £ ( a 3 + «5), bt=~ T2 (a- + a 7 ), & 6 ==^(a 7 + a 9 ), etc. ; 



thus, for instance, & 2 = 43/50, Z> 4 =191/490 ; the following 

 ones are far from claiming to be elegant numbers. For 

 purposes of practical calculation the logarithms of these 

 coefficients will be most convenient ; these, enabling the 

 reader to use (19 a) up to w s , are 



log b 2 = 1-93450, log I u = 1-59083,1 



_ I . (21) 



log b Q = 2-68220, log & 8 = 3-48187, J 



They will be found sufficient up to iv = 1*1 or 1*2, and 

 beyond that the original form (19), with the upper line 

 of (17), must be used, the power series then ceasing to 

 converge. 



It may be mentioned here that a direct comparison o£ 

 the radiation formula (14) with Lummer and Pringsheim's 



