Radiation on the Transmission of Heat. 245 



Hence, taking account of (1), 



i^~ B ^-4e ( 5 > 



From (4) and (5) we may deduce 



£ta+b)=.£ ( b-a) 



du 

 - KC d6 ; 



and finally by two differentiations of (1) and elimination of 

 A and B, 



dr_ „\ du . n d 2 F 

 \dx 



^-" 2 >i.+^=°- -■■ ■ «0 



If the variations of temperature are sufficiently small to 

 allow Newton's law to be applied, we may write F = R» ; 

 so that 



/ d 2 A du , n ^d 2 ^ n 



U?-"T^ + 2kR ^ =0 - • • • (7) 



This is in essence the equation which Prof. Sampson has 

 deduced in a different manner. Though I am not in agree- 

 ment with Prof. Sampson as regards the deductions he has 

 made from it, the priority of having established a somewhat 

 important equation belongs to him. 



If we adopt Stefan's law, we must write F = Si« 4 ^ so that we 

 obtain the more correct equation 



/ d 2 A du , d 2 u* A 



It will be noticed that the equation remains unaltered when 

 cp, Kp, pdx is writen for c, k, dx respectively ; so that we 

 may take c to denote the specific heat, tc the coefficient of 

 absorption per unit mass of a thin sheet, and dx the mass 

 per unit surface of the sheet. 



3. To consider more particularly the case of steady tempe- 

 rature, I imagine a plate of a non-conducting and partially 

 transparent material placed with its surfaces in contact with 

 two other surfaces which are kept at constant but different 

 temperatures. These surfaces will radiate heat: and if they 

 are black the radiation towards the inside of the plate is known 

 at both boundaries. The surface conditions are (1) Eor *=0, 

 A = F , and (2) for x = d, B = F l5 where F and ¥ x are the 



