482 THEORY OF HEAT. [CHAP. IX. 



we need only add that to obtain a profound conviction of the use 

 fulness of these principles it is necessary to consider also various 

 difficult problems; for example, that which we are about to in 

 dicate, and whose solution is wanting to our theory, as we have 

 long since remarked. This problem consists in forming the differ 

 ential equations, which express the distribution of heat in fluids 

 in motion, when all the molecules are displaced by any forces, 

 combined with the changes of temperature. The equations which 

 we gave in the course of the year 1820 belong to general hydro 

 dynamics; they complete this branch of analytical mechanics 1 . 



430. Different bodies enjoy very unequally the property which 

 physicists have called conductibility or conducibility , that is to say, 

 the faculty of admitting heat, or of propagating it in the interior 

 of their masses. We have not changed these names, though they 



1 See Memoires de V Academic des Sciences, Tome xn. Paris, 1833, pp. 515530. 



In addition to the three ordinary equations of motion of an incompressible 

 fluid, and the equation of continuity referred to rectangular axes in direction of 

 which the velocities of a molecule passing the point x, y, z at time t are u, v, w, 

 its temperature being 6, Fourier has obtained the equation 



in which K is the conductivity and C the specific heat per unit volume, as 

 follows. 



Into the parallelepiped whose opposite corners are (x, y, z), (x + Ax,y + Ay, z + Az), 

 the quantity of heat which would flow by conduction across the lower face AxAy, 



if the fluid were at rest, would be -K-j- AxAy At in time At, and the gain by 

 convection + Cw Ax Ay At ; there is a corresponding loss at the upper face Ax Ay ; 

 hence the whole gain is, negatively, the variation of (-K~,~+ Cwd) Ax Ay At with 



respect to z, that is to say, the gain is equal to ( K -^ - C - - (w0) } Ax Ay Az At. 

 Two similar expressions denote the gains in direction of y and z ; the sum of the 

 three is equal to (7 At Ax Ay Az, which is the gain in the volume Ax Ay Az 



in time At : whence the above equation. 



The coefficients K and C vary with the temperature and pressure but are 

 usually treated as constant. The density, even for fluids denominated incom 

 pressible, is subject to a small temperature variation. 



It may be noticed that when the velocities u, v, w are nul, the equation 

 reduces to the equation for flow of heat in a solid. 



It may also be remarked that when K is so small as to be negligible, the 

 equation has the same form as the equation of continuity. [A. F.j 



