Mr Wilson, On Convection of Heat. 417 



which is Fourier's solution for the corresponding problem in a slab 

 at rest. 



As an example, suppose that where x is negative the planes 

 A B and CD are maintained at the temperature unity and where 

 x is positive at the temperature zero, and let there be no sources 

 of heat along Oy. For simplicity also let t — ir. 



Then, when x is negative, the solution is 



= 1+2 A n e (p+ ^ +]?i)x sinm/, 



n=X 



and, when x is positive, it is 



6= 2 B n e {p '^ n2+P ' 2)x smny. 



n = X 



It remains therefore to determine the A's and the B's. When 

 x = 0, we have 



1 + "ZA n sin ny = %B n sin ny. 



Hence X (B n — A n ) sin ny = 1. 



4 



Consequently B n — A n = — when n is odd and zero when n is 



nir 



even. 



Also, when x = 0, 



j a 



-T- = %A n (p + Jn 2 + p 2 ) sin ny = %B n (p - Jn 2 + p 2 ) sin ny. 



This equation is satisfied if 



_ p-Jn 2 +p 2 

 p + sjn 2 +p- 

 Therefore, when n is odd, 



Hence B n = — ( 1 + 



p + n/w 2 + pV nir 



2 A , i 5 



727T \ 



Jn 2 + ja 2 / 

 and, when n is even, .£,,, = 0. Also, when n is odd, 



A ~ 2 f p 



m 

 and, when n is even, J.„ = 0. 



A n — Bn, — , 



nir mr\J n *+p* 



