SECT. VI.] GENERAL EQUATIONS OF PROPAGATION. 107 



tained, one at the temperature g, and the other at temperature 0, 

 the constant flow of heat, in this second hypothesis, or the quantity 

 which during unit of time crosses unit of surface taken on an 

 intermediate plane parallel to the bases, is equal to the product 

 of the first flow multiplied by g. 



In fact, since all the temperatures have been increased in 

 the ratio of 1 to g, the differences of the temperatures of any 

 two points whatever m and //., are increased in the same ratio. 

 Hence, according to the principle of the communication of heat, 

 in order to ascertain the quantity of heat which in sends to ^ 

 on the second hypothesis, we must multiply by g the quantity 

 which the same point m sends to (JL on the first hypothesis. 

 The same would be true for any two other points whatever. 

 Now, the quantity of heat which crosses a plane M results from 

 the sum of all the actions which the points m, m , m&quot;j m&quot;, etc., 

 situated on the same side of the plane, exert on the points //., 

 //, fju , fj!&quot; } etc., situated on the other side. Hence, if in the first 

 hypothesis the constant flow is denoted by K } it will be equal to 

 gK, w r hen we have multiplied all the temperatures by g. 



137. THEOREM II. In a prism whose constant temperatures 

 are expressed by the equation v = A ax- by cz, and which 

 is bounded by six planes at right angles all of whose points are 

 maintained at constant temperatures determined by the preceding 

 equation, the quantity of heat which, during unit of time, crosses 

 unit of surface taken on any intermediate plane whatever perpen 

 dicular to z, is the same as the constant flow in a solid of the 

 same substance would be, if enclosed between two infinite parallel 

 planes, and for which the equation of constant temperatures is 

 v = c cz. 



To prove this, let us consider in the prism, and also in the 

 infinite solid, two extremely near points m and p, separated 



Fig. 4. 



r 



m h 



by the plane M perpendicular to the axis of z ; ^ being above 

 the plane, and m below it (see fig. 4), and above the same plane 



