74 THEORY OF HEAT. [CHAP. I. 



we shall prove that the same causes which, by hypothesis, keep 

 the outer layers of the solid in their initial state, are sufficient 

 to preserve also the actual temperatures of every one of the inner 

 molecules, so that their temperatures do not cease to be repre 

 sented by the linear equation. 



The examination of this question is an element of the 

 general theory, it will serve to determine the laws of the varied 

 movement of heat in the interior of a solid of any form whatever, 

 for every one of the prismatic molecules of which the body is 

 composed is during an infinitely small time in a state similar 

 to that which the linear equation (a) expresses. We may then, 

 by following the ordinary principles of the differential calculus, 

 easily deduce from the notion of uniform movement the general 

 equations of varied movement. 



93. In order to prove that when the extreme layers of the 

 solid preserve their temperatures no change can happen in the 

 interior of the mass, it is sufficient to compare with each other 

 the quantities of heat which, during the same instant, cross two 

 parallel planes. 



Let b be the perpendicular distance of these two planes which 

 we first suppose parallel to the horizontal plane of x and y. Let 

 m and m be two infinitely near molecules, one of which is above 

 the first horizontal plane and the other below it : let x, y, z be 

 the co-ordinates of the first molecule, and x, y f , z those of the 

 second. In like manner let M and M denote two infinitely 

 near molecules, separated by the second horizontal plane and 

 situated, relatively to that plane, in the same manner as m and 

 m are relatively to the first plane ; that is to say, the co-ordinates 

 of M are a?, y, z + b, and those of M are x, y , z + b. It is evident 

 that the distance mm of the two molecules m and mf is equal 

 to the distance MM of the two molecules M and M f ; further, 

 let v be the temperature of m, and v that of m, also let V and 

 V be the temperatures of M and M f , it is easy to see that the 

 two differences v v and V V are equal ; in fact, substituting 

 first the co-ordinates of m and m in the general equation 



v A + ax -f by + cz, 

 we find v v = a (x - x) -f b (y y} + c (z z}, 



