76 THEORY OF HEAT. [CHAP. I. 



planes and maintained in a constant state. We have seen that 

 the temperatures of the different points of the mass are in this 

 case represented by the equation v A + cz ; we proceed to prove 

 that the uniform flow of heat propagated in the vertical direction 

 in the infinite solid is equal to that which flows in the same 

 direction across the prism enclosed by six planes at right angles. 

 This equality necessarily exists if the coefficient c in the equation 

 v = A + cz, belonging to the first solid, is the same as the coeffi 

 cient c in the more general equation v A + ax + ~by + cz which 

 represents the state of the prism. In fact, denoting by H a 

 plane in this prism perpendicular to z t and by m and /JL two 

 molecules very near to each other, the first of which m is below 

 the plane H, and the second above this plane, let v be the 

 temperature of m whose co-ordinates are x, y, z, and w the 

 temperature of //, whose co-ordinates are x -H a, y + /3. z + 7. Take 

 a third molecule fi whose co-ordinates are x a., y /3, # + y, and 

 whose temperature may be denoted by w. We see that fju and 

 fju are on the same horizontal plane, and that the vertical drawn 

 from the middle point of the line fjup , which joins these two 

 points, passes through the point m, so that the distances mjj, and 

 mfjf are equal. The action of m on ^ or the quantity of heat 

 which the first of these molecules sends to the other across the 

 plane H, depends on the difference v - w of their temperatures. 

 The action of m on p depends in the same manner on the 

 difference v w of the temperatures of these molecules, since 

 the distance of m from fju is the same as that of m from /* . Thus, 

 expressing by q (v w) the action of m on //, during the unit of 

 time, we shall have q (v w) to express the action of m on fjf, 

 q being a common unknown factor, depending on the distance 

 nifjb and on the nature of the solid. Hence the sum of the two 

 actions exerted during unit of time is q (v w + v w }. 

 If instead of x, y, and z t in the general equation 



v = A + ax + by + cz, 



we substitute the co-ordinates of m and then those of p and //, 

 we shall find 



t? w = act 6/3 c% 



v w = + ay. + bft cy. 



