Mr Wilson, On Convection of Heat. 421 



If the condition given is that no heat shall pass through the 

 planes AB and CD, then we must have ^- = when y = and 

 when y = t. In this case we take a solution of the form 



71 = V t J 



so that -j- = 2 - ii n — e a "* sm - , 



ay «=o < V t / 



which is zero when y = or t, and so satisfies the given condition. 



We shall now consider a problem to which the solution just 

 obtained is not applicable. Let the temperature of the planes 

 AB and CD be given by 6 = ax + b, where a and b are constants. 



In this case it is clear that j-^ = and -=-=a everywhere 

 in the slab. The differential equation therefore becomes 



SOU 



where p = -^ as before. The solution of this is 



= pay 2 + Cay + D, 

 where C and D are constants. When y = 0, 



6 = D = ax + b, 

 and when y = t, 6 = pat 2 + Cat + ax + b = ax + b, 



hence C = —pt. The solution of the problem is therefore 

 6 — pay 2 — paty + ax + b. 



In this case therefore the isothermal lines are parabolas with 



9, 

 is given by 



their axes at y = — . The temperature at the centre of the slab 



or = a# + b — "^-t— , 



4 



so that the centre of the slab is cooler than the sides by ^-r 



