SECT. IV.] UNIFORM LINEAR MOVEMENT. 55 



From this may be derived a /3= p j-~ an equation 



fl6 ~\~ K 



whose second member is known ; for the temperatures a and 6 

 are given, as are also the quantities h, ^, e. 



Introducing this value of a- ft into the general equation 



v = a + - z, we shall have, to express the temperatures of any 

 section of the solid, the equation a v=-^~ j - , in which 



llG ~r~ rC 



known quantities only enter with the corresponding variables v 

 and z. 



72. So far we have determined the final and permanent state 

 of the temperatures in a solid enclosed between two infinite and 

 parallel plane surfaces, maintained at unequal temperatures. 

 This first case is, properly speaking, the case of the linear and 

 uniform propagation of heat, for there is no transfer of heat in 

 the plane parallel to the sides of the solid ; that which traverses 

 the solid flaws uniformly, since the value of the flow is the same 

 for all instants and for all sections. 



We will now restate the three chief propositions which result 

 from the examination of this problem ; they are susceptible of a 

 great number of applications, and form the first elements of our 

 theory. 



1st. If at the two extremities of the thickness e of the solid 

 we erect perpendiculars to represent the temperatures a and b 

 of the two sides, and if we draw the straight line which joins 

 the extremities of these two first ordinates, all the intermediate 

 temperatures will be proportional to the ordinates of this straight 



line ; they are expressed by the general equation a v = - - z, 



6 



v denoting the temperature of the section whose height is z. 



2nd. The quantity of heat which flows uniformly, during 

 unit of time, across unit of surface taken on any section whatever 

 parallel to the sides, all other things being equal, is directly 

 proportional to the difference a b of the extreme temperatures, 

 and inversely proportional to the distance e which separates 



^a-6 

 these sides. The quantity of heat is expressed by K - , or 



