94 THEORY OF HEAT. [CHAT*. II. 



receives when we give to x its complete value X\ hence we obtain 



the equation -,- = -jyV, which must hold when in the functions 

 dx A 



Ct ?J 



T and v we put instead of x its value X, which we shall denote 

 dx 



dV 



by writing it in the form K ~j- + h V 0. 



doc 



116. The value of -=- taken when x = X, must therefore have 



dx 



a constant ratio -+ to the value of v, which corresponds to the 



same point. Thus we shall suppose that the external cause of 

 the cooling determines always the state of the very thin envelope, 



C/1J 



in such a manner that the value of , -- which results from this 



dx 



state, is proportional to the value of v, corresponding to x = X, 

 and that the constant ratio of these two quantities is -^ . This 

 condition being fulfilled by means of some cause always present, 

 which prevents the extreme value of -y- from being anything else 



CLX 



but ^ v, the action of the envelope will take the place of that 



of the air. 



It is not necessary to suppose the envelope to be extremely 

 thin, and it will be seen in the sequel that it may have an 

 indefinite thickness. Here the thickness is considered to be 

 indefinitely small, so as to fix the attention on the state of the 

 surface only of the solid. 



117. Hence it follows that the three equations which are 

 required to determine the function $ (x, t} or v are the following, 



dn 



Tt~~ 



The first applies to all possible values of x and t ; the second 

 is satisfied when x = X, whatever be the value of t; and the 

 third is satisfied when t = 0, whatever be the value of x. 



