ON THE CONDUCTION OF HEAT. 13 



The solution of the former is 



which, when substituted in the latter equation, gives for the extremity of the 

 bar whose length is b, 



Be ^ ki — Ke ^ ki t^^ 



V'ik + Vhl 



and 



whena; = 0, M = A-^ 1 H -= :^^-=e v */ > 



.'. A = E(V2lt- V7ri)u; 

 abbreviating the reciprocal of V^k + ^hl 



, 91 /2A 



^{^/^lk~ Vhl)e ^V*7byE. 

 Hence 



When x=.b, v' = C ti. 



11. Cor. The temperature at any point of a bar of finite length is repre- 

 sented by 



The expression shows that the temperature is the difference of two quan- 

 tities ; the one due to the heating at the one extremity, and the other follow- 

 ing a corresponding law, and due to the cooling at the other extremity of the 

 bar. 



12. On Libri's hypothesis, the expression for the temperature at the extre- 

 mity of a bar heated permanently at the other extremity is 



"o 



where u is the temperature at the heated extremity of the bar, and p, q, p 

 q , are constants. This proposition is of some importance, but from the 

 want of experiments we do not think it necessary to exhibit the values of the 

 constants in full. 



The demonstration is as follows : since 



at the extremity of the bar, we obtain 



6 



. B-log,a -2o(b-x) , „, 

 + 2£_e "^^ ^ — ABlog^a. 



