Radiation on the Transmission of Heat. 249 



The solution of (10) may be put into the form : 



u^e^+Qe-^ + ax + b. .... (12) 



P, Q, a, and b are four constants, and a satisfies the equation 



« 2 \.= 2a:R + ^X. . . , . . . (13) 



The constants are determined from our knowledge of u b and 

 Ud at the ends of the plate and the conditions which hold then* 

 as regards radiation. 



The heat (H) transmitted in unit time through any 

 isothermal plane in the negative direction is 



H=\'J" +B-A. 

 ax 



From the original equation 



d(A-'rB) 



dx 



tf(B-A), 



and 



«(A-+B)=2«F-\^? ; (14) 



ax" ' 



••• K '( B - A ) = - K (te -\^ ( 15 > 



Substituting F = R^, the heat transmitted takes the form 



, T __ \/ 9 du dhi\ 



k 2 \ dx doc z J 



From equation (17) we deduce 



dhi 

 dx* 



'(=-)■ 



so that finally for the heat transmitted we have the simple 

 expressions 



H = a— - r = ( — +\)a (16 



AT \ K, J 



In this equation R is a constant such that the excess of 

 heat radiated from unit area of an infinite plane black surface 

 of temperature iii over that received from a similar surface 

 placed parallel to it and of temperature u Q is R(ux— 1* ) : k is 

 a quantity such that the radiation falling on a plate of thickness 

 l//c is by absorption within the plate reduced in the ratio < : 1 ; 

 A, is the thermal conductivity, and a is a quantity which must 

 be determined from the conditions of the problem. 



If ~U=u f —u be the difference in temperature between the 



