SECT. V.] STEADY TEMPERATURE IN A BAR. 57 



tained at a constant temperature 0, and carried away by a 

 current with uniform velocity. 



Within the interior of the solid, heat will pass successively 

 all the parts situate to the right^of the source, and not exposed 

 directly to its action; they will be heated more and more, but 

 the temperature of each point will not increase beyond a certain 

 limit. This maximum temperature is not the same for every 

 section ; it in general decreases as the distance of the section 

 from the origin increases : we shall denote by v the fixed tem 

 perature of a section perpendicular to the axis, and situate at a 

 distance x from the origin A 



Before every point of the solid has attained its highest degree 

 of heat, the system of temperatures varies continually, and ap 

 proaches more and more to a fixed state, which is that which 

 we consider. This final state is kept up of itself when it has 

 once been formed. In order that the system of temperatures 

 may be permanent, it is necessary that the quantity of heat 

 which, during unit of time, crosses a section made at a distance x 

 from the origin, should balance exactly all the heat which, during 

 the same time, escapes through that part of the external surface 

 of the prism which is situated to the right of the same section. 

 The lamina whose thickness is dx, and whose external surface 

 is Sldx, allows the escape into the air, during unit of time, of 

 a quantity of beat expressed by Shlv . dx, h being the measure of 

 the external conducibility of the prism. Hence taking the in 

 tegral jShlv . dx from x = to x oo , we shall find the quantity 

 of heat w r hich escapes from the whole surface of the bar durino- 

 unit of time ; and if we take the same integral from x = to 

 x = x, we shall have the quantity of heat lost through the part 

 of the surface included between the source of heat and the section 

 made at the distance x. Denoting the first integral by (7, whose 

 value is constant, and the variable value of the second by 

 jShlv.dx-, the difference C-/8hlv.dx will express the whole 

 quantity of heat which escapes into the air across the part of 

 the surface situate to the right of the section. On the other 

 hand, the lamina of the solid, enclosed between two sections 

 infinitely near at distances x and x + dx, must resemble an in 

 finite solid, bounded by two parallel planes, subject to fixed 

 temperatures v and v + dv, since, by hypothesis, the temperature 



