108 THEORY OF HEAT. [CHAP. II. 



let us take a point m such that the perpendicular dropped from 

 the point //, on the plane may also be perpendicular to the 

 distance mm at its middle point h. Denote by x, y, z + h, the 

 co-ordinates of the point //,, whose temperature is w, by x a, y /3, 

 z, the co-ordinates of m, whose temperature is v, and by a? -fa, 

 y + {3, z, the co-ordinates of m , whose temperature is v. 



The action of m on (JL, or the quantity of heat which m sends 

 to jju during a certain time, may be expressed by q(v w). The 

 factor q depends on the distance nip, and on the nature of the 

 mass. The action of m on //, will therefore be expressed by 

 q (v w) ; and the factor q is the same as in the preceding 

 expression; hence the sum of the two actions of m on ft, and 

 of m on //-, or the quantity of heat which //, receives from m and 

 from m, is expressed by 



q ( v w -f v w}. 

 Now, if the points m, p, m belong to the prism, we have 



w A ax by c (z -f h), v = A a (x a) b (y /3) cz, 

 and v = A - a (x + a) - 6 (y + /3) - cz ; 



and if the same points belonged to an infinite solid, we should 

 have, by hypothesis, 



w = c c(z+li) y v = c cz, and v = c cz. 

 In the first case, we find 



q (v w + v w) = 2qch, 



and, in the second case, we still have the same result. Hence 

 the quantity of heat which //, receives from m and from m on 

 the first hypothesis, when the equation of constant temperatures 

 is v = A ax by cz, is equivalent to the quantity of heat 

 which p receives from m and from m when the equation of 

 constant temperatures is v = c cz. 



The same conclusion might be drawn with respect to any three 

 other points whatever m, /// , m&quot;, provided that the second // be 

 placed at equal distances from the other two, and the altitude of 

 the isosceles triangle m /jf m&quot; be parallel to z. Now, the quantity 

 of heat which crosses any plane whatever M, results from the sum 

 of the actions which all the points m, m , in&quot;, in&quot; etc., situated on 



