110 THEORY OF HEAT. [CHAP. II. 



Let w be an infinitely small circle whose centre coincides with 

 the point m of the solid, and whose plane is perpendicular to the 

 vertical co-ordinate z ; during the instant dt there will flow across 

 this circle a certain quantity of heat which will pass from the 

 part of the circle below the plane of the circle into the upper 

 part. This flow is composed of all the rays of heat which depart 

 from a lower point arid arrive at an upper point, by crossing 

 a point of the small surface w. We proceed to shew that the 



dv 

 expression of the value of the flow is K -7- &&amp;gt;dt. 



Let us denote by x, y, z the coordinates of the point m whose 

 temperature is v ; and suppose all the other molecules to be 

 referred to this point in chosen as the origin of new axes parallel 

 to the former axes : let f, 77, f, be the three co-ordinates of a point 

 referred to the origin m ; in order to express the actual temperature 

 w of a molecule infinitely near to m, we shall have the linear 

 equation 



, ,. dv dv . dv 

 w-v + -r+i7-7 +-,-. 

 * dx dy dz 



The coefficients t/, j-n. -7, -r- are the values which are found 

 dx dy dz 



by substituting in the functions v,j-, -j- , -T-, for the variables 



x, y z, the constant quantities x r , y, z, which measure the dis 

 tances of the point m from the first three axes of x, y, and z. 



Suppose now that the point m is also an internal molecule of 

 a rectangular prism, enclosed between six planes perpendicular to 

 the three axes whose origin is m ; that w the actual temperature of 

 each molecule of this prism, whose dimensions are finite, is ex 

 pressed by the linear equation w = A + a% + brj + c and that the 

 six faces which bound the prism are maintained at the fixed tem 

 peratures which the last equation assigns to them. The state of 

 the internal molecules will also be permanent, and a quantity of 

 heat measured by the expression Kcwdt will flow during the 

 instant dt across the circle &&amp;gt;. 



This arranged, if we take as the values of the constants 



7 xi ,-,- dv dv dv ,, / j c ,1 



A, a, 6, c, the quantities v , -5 , -y- , -j- t the fixed state of the 



