250 Prof. A. Schuster on the Influence of 



two sides of the plate at which x is t and respectively, 

 equation (12) gives 



JJ = -p(e af ,-l)+Q{e- at -l)+at. . . . (17) 



Two further equations are required to determine the con- 

 stants P, Q, and a. These are obtained by considering the 

 relations which hold between A and B at the two surfaces. The 

 radiation A leaving the surface x=0 is made up partly by 

 the reflexion of the incident radiation B which we may put 

 equal to o-B, and partly by the radiation of the surface itself 

 which is (1 — a)B.u. Hence A , B , u , the values of A, B. u 

 for .v = 0, 



A — o\B =(l — a)B.u (18a) 



Similarly at the second surface where ,v = t 



B*- </A t = (l~o-0 B,u t . . . . (186) 



For surfaces which are black cr = 0, for surfaces which are 

 totally reflecting cr = l. Substituting the values of F and u 

 in (11) and (15) we find for x = and x = t 



*(B + A ) = 2*R« -\a 2 (P + Q), 



f c(B -A )=2aR-*K\(P-Q), | 



k. . . . (ly) 

 tc(B f + At) ^2KB.u t -Xu 2 (Ve at 4-Qe- at ) [ 



*(B*— A*) =2aR-xK\(Fe* t ~Q,e- at ) j 



The four last equations determine A , B , A t , B f in terms of 

 a, P, Q, and hence by substitution in (18 a) and (18 b) the two 

 relations which together with (17) allow the three constants 

 of the equation to be calculated. The calculation of these 

 constants is simplified when a = a', so that the two surfaces 

 have equal reflecting properties. Symmetry shows in that 

 case that the temperature at the centre of the plate must be 

 i(u + Ut). Applying (12) we obtain the equation 



i(ut + u )=im + ^)+Q(l + e-^) + ^+b, 



and for u— - a l ~ - ?/ 



2 u — Ye^ + Qe 2 +b. 



Equating the two expressions w T e derive 



P^ + Q=0 (20) 



Hence (17) becomes 



U=2P(«f— l)+a* (21) 



Only one further equation is necessary, which is obtained 



