56 THEORY OF HEAT. [CHAP. I. 



K , if we derive from the general equation the value of 



-v- which is constant; this uniform flow may always be repre 

 sented, for a given substance and in the solid under examination, 

 by the tangent of the angle included between the perpendicular 

 e and the straight line whose ordinates represent the tempera 

 tures. 



3rd. One of the extreme surfaces of the solid being submitted 

 always to the temperature a, if the other plane is exposed to air 

 maintained at a fixed temperature b ; the plane in contact with 

 the air acquires, as in the preceding case, a fixed temperature /?, 

 greater than b, and it permits a quantity of heat to escape into 

 the air across unit of surface, during unit of time, which is ex 

 pressed by h (/3 b) , h denoting the external conducibility of 

 the plane. 



The same flow of heat h(/3 b) is equal to that which 

 traverses the prism and whose value is K(a ft)\ we have there 

 fore the equation h({3 ft) = K , which gives the value 

 of 



SECTION V. 



Law of the permanent temperatures in a prism of small 



thickness. 



73. We shall easily apply the principles which have just 

 been explained to the following problem, very simple in itself, 

 but one whose solution it is important to base on exact theory. 



A metal bar, whose form is that of a rectangular parallele 

 piped infinite in length, is exposed to the action of a source of 

 heat which produces a constant temperature at all points of its 

 extremity A. It is required to determine the fixed temperatures 

 at the different sections of the bar. 



The section perpendicular to the axis is supposed to be a 

 square whose side 21 is so small that we may without sensible 

 error consider the temperatures to be equal at different points 

 of the same section. The air in which the bar is placed is main- 



