116 THEORY OF HEAT. [CHAP. II. 



value of v as a function of the four variables x, y, z, t. Differen 

 tiating the equation f(x, y, z) = 0, we shall have 



mdx 4- ndy -\-pdz ; 



m, n, p being functions of x, y, z. 



It follows from the corollary enunciated in Article 141, that 

 the flow in direction of the normal, or the quantity of heat which 

 during the instant dt would cross the surface , if it were placed 

 at any point whatever of this line, at right angles to its direction, 

 is proportional to the quotient which is obtained by dividing the 

 difference of temperature of two points infinitely near by their 

 distance. Hence the expression for the flow at the end of the 

 normal is 



T ^w v T 

 K - codt] 



GC 



K denoting the specific conducibility of the mass. On the other 

 hand, the surface co permits a quantity of heat to escape into the 

 air, during the time dt, equal to hvcodt ; h being the conducibility 

 relative to atmospheric air. Thus the flow of heat at the end of 

 the normal has two different expressions, that is to say : 



hvcodt and K - codt ; 



hence these two quantities are equal ; and it is by the expression 

 of this equality that the condition relative to the surface is in 

 troduced into the analysis. 



147. We have 



, . dv ^ dv ~ dv 

 w v + ov = v + -y- ox + -j- oy -f- -j~ oz. 

 ax dy dz 



Now, it follows from the principles of geometry, that the co 

 ordinates $x, &/, &z, which fix the position of the point v of the 

 normal relative to the point ^ satisfy the following conditions : 



We have therefore 



w 



1 / dv dv dv\ &amp;lt;* 

 -v = - (m-j- + n-j- + p^-) oz: 

 p\ dx dy * dz 



