112 THEORY OF HEAT. [CHAP. II. 



fixed point o, and by v the ordinate which represents the tem 

 perature of the point p, v will vary with the distance e and 

 will be a certain function /(e) of that distance; the quantity 

 of heat which would flow across the circle o&amp;gt;, placed at the 



point p perpendicular to the line, will be K -=- wdt, or 



-Kf (e)a&amp;gt;dt, 

 denoting the function \/ by/ (e). 



QJ. 



We may express this result in the following manner, which 

 facilitates its application. 



To obtain the actual flow of heat at a point p of a straight 

 line drawn in a solid, whose temperatures vary by action of the 

 molecules, we must divide the difference of the temperatures at 

 two points infinitely near to the point p by the distance between 

 these points. The flow is proportional to the quotient. 



142. THEOHEM IV. From the preceding Theorems it is 

 easy to deduce the general equations of the propagation of heat. 



Suppose the different points of a homogeneous solid of any 

 form whatever, to have received initial temperatures which vary 

 successively by the effect of the mutual action of the molecules, 

 and suppose the equation v = f (x, y, z, t) to represent the successive 

 states of the solid, it may now be shewn that v a function of four 

 variables necessarily satisfies the equation 



dy K_ /d 2 v dV dV\ 

 dt &quot; CD Vdx 2 + dy* + dzV 



In fact, let us consider the movement of heat in a molecule 

 enclosed between six planes at right angles to the axes of x, y, 

 and z\ the first three of these planes pass through the point 

 m whose coordinates are x, y, z, the other three pass through 

 the point m, whose coordinates are x + dx, y + dy,z + dz. 



During the instant dt, the molecule receives, across the 

 lower rectangle dxdy, which passes through the point m, a 



quantity of heat equal to K dx dy -=- dt. To obtain the quantity 



which escapes from the molecule by the opposite face, it is 

 sufficient to change z into z -f dz in the preceding expression, 



