248 Prof. A. Schuster on the Influence of 



For perfect reflectors <7=1, and the discontinuities are 



i(Fi-Fo); 



and hence the more perfectly the boundaries reflect the more 

 nearly will the plate assume a uniform temperature, which is 

 halfway between the temperatures of the bounding surfaces. 

 5. If conduction be taken into account, equation (1) 

 becomes 



k(A + B-2F) + \^=c^, .... (9) 



X being thermal conductivity of the body. 



Differentiating twice and substituting equations (4«) and 

 (46), 



= /c s (2F-A-B). 



This equation and (9) allow A + B to be eliminated, giving 

 the differential equation 



d*F / « 2 9 W du <Pu\ 



When the temperature is steady and its variations across 

 the plate are sufficiently small to admit of the interchange 

 of radiation between two bodies to be proportional to the 

 difference in their temperatures, the equation becomes 





,_ -p. a ^ ,d Q u „ dSt 



or. say, 







2 d*u d 4 u 



(11) 



The distribution of temperature in the plate is here not a 

 linear function of the distance. This is a curious result. 

 Radiation alone or conduction alone would cause the slope 

 of temperature in the plate to be uniform ; but the combi- 

 nation of both effects destroys the uniformity. The explanation 

 of the apparent paradox is found in the consideration that 

 although the distribution of temperature due to radiation 

 varies uniformly with the distance across the plate, it is 

 not identical with the distribution of temperature due to 

 conduction, owing to the tendency of radiation to cause dis- 

 continuity at the surfaces, as has been explained in § 3. 



