80 THEORY OF HEAT. [CHAP. I. 



The expression of the quantity of heat which, during the in 

 stant dt, flows across a horizontal circle infinitely small, whose area 

 is &&amp;gt;, and passes in this manner from the part of the solid which is 

 below the plane of the circle to the part above, is, for the two cases 



rr dv j, 

 in question, K -^- coat. 



98. It is easy now to generalise this result and to recognise 

 that it exists in every case of the varied movement of heat ex 

 pressed by the equation v = F (x, y, z, t). 



Let us in fact denote by x, y, z , the co-ordinates of this point 

 m, and its actual temperature by v. Let x + f, y + rj, z -f f, be 

 the co-ordinates of a point JJL infinitely near to the point m, and 

 whose temperature is w ; f, r\, are quantities infinitely small added 

 to the co-ordinates x , y , z ; they determine the position of 

 molecules infinitely near to the point m, with respect to three 

 rectangular axes, whose origin is at m, parallel to the axes of 

 x, y, and z. Differentiating the equation 



=/ 0&amp;gt; y&amp;gt; z &amp;gt; 



and replacing the differentials by f, rj, we shall have, to express 

 the value of w which is equivalent to v + dv, the linear equation 



, dv ,. dv dv ^ , m . , dv dv dv f 

 w = v + j- f + ~j- v + -7- ? ; the coefficients v , -y-, --,-, i- , are func- 

 dx dy dz . dx dy dz 



tions of x, y, z, t, in which the given and constant values of, y } z, 

 which belong to the point m, have been substituted for x, y&amp;gt; z. 



Suppose that the same point m belongs also to a solid enclosed 

 between six rectangular planes, and that the actual temperatures 

 of the points of this prism, whose dimensions are finite, are ex 

 pressed by the linear equation w = A + a + Irj + c ; and that 

 the molecules situated on the faces which bound the solid are 

 maintained by some external cause at the temperature which is 

 assigned to them by the linear equation, f, rj, are the rectangular 

 co-ordinates of a molecule of the prism, whose temperature is w t 

 referred to three axes whose origin is at m. 



This arranged, if we take as the values of the constant coeffi 

 cients A, a, 6, c, which enter into the equation for the prism^ the 



,.,. , dv dv dv r , . , , , ,. ,,-p &amp;lt;.. 



quantities v , -y- , -=- , -=- , which belong to the ditierential eqna- 

 cLoc dy cLz 



tion ; the state of the prism expressed by the equation 



