82 THEORY OF HEAT. [CHAP. I. 



The fluxion of the ordinate of this curve, answering to the point 

 ra, taken with the opposite sign, expresses the velocity with 

 which heat is transferred across the plane. This fluxion of the 

 ordinate is known to be the tangent of the angle formed by 

 the element of the curve with a parallel to the abscissse. 



The result which we have just explained is that of which 

 the most frequent applications have been made in the theory 

 of heat. We cannot discuss the different problems without 

 forming a very exact idea of the value of the flow at every point 

 of a body whose temperatures are variable. It is necessary to 

 insist on this fundamental notion ; an example which we are 

 about to refer to will indicate more clearly the use which has 

 been made of it in analysis. 



100. Suppose the different points of a cubic mass, an edge 

 of which has the length TT, to have unequal actual temperatures 

 represented by the equation v = cos x cos y cos z. The co 

 ordinates x, y, z are measured on three rectangular axes, whose 

 origin is at the centre of the cube, perpendicular to the faces. 

 The points of the external surface of the solid are at the actual 

 temperature 0, and it is supposed also that external causes 

 maintain at all these points the actual temperature 0. On this 

 hypothesis the body will be cooled more and more, the tem 

 peratures of all the points situated in the interior of the mass 

 will vary, and, after an infinite time, they will all attain the 

 temperature of the surface. Now, we shall prove in the sequel, 

 that the variable state of this solid is expressed by the equation 



v = e~ 9t cos x cos y cos z, 



3/iT 



the coefficient g is equal to * 71 -^ * s ^ ne specific conduci- 



G . I) 



bility of the substance of which the solid is formed, D is the 

 density and G the specific heat ; t is the time elapsed. 



We here suppose that the truth of this equation is admitted, 

 and we proceed to examine the use which may be made of it 

 to find the quantity of heat which crosses a given plane parallel 

 to one of the three planes at the right angles. 



If, through the point m, whose co-ordinates are x, y, z, we 

 draw a plane perpendicular to z, we shall find, after the mode 



