18 THEORY OF HEAT. [CHAP. I. 



which flow out at each instant through the whole external surface 

 of the solid, are variable and comparable with each other ; their 

 ratios are analytical functions depending on the time t, the expres 

 sion of which must be assigned. 



9. Let us examine also the case in which a rectangular prism 

 of sufficiently great thickness and of infinite length, being sub 

 mitted at its extremity to a constant temperature, whilst the air 

 which surrounds it is maintained at a less temperature, has at last 

 arrived at a fixed state which it is required to determine. All the 

 points of the extreme section at the base of the prism have, by 

 hypothesis, a common and permanent temperature. It is not the 

 same with a section distant from the source of heat; each of the 

 points of this rectangular surface parallel to the base has acquired 

 a fixed temperature, but this is not the same at different points of 

 the same section, and must be less at points nearer to the surface 

 exposed to the air. We see also that, at each instant, there flows 

 across a given section a certain quantity of heat, which always 

 remains the same, since the state of the solid has become constant. 

 The problem consists in determining the permanent temperature 

 at any given point of the solid, and the whole quantity of heat 

 which, in a definite time, flows across a section whose position is 

 given. 



10. Take as origin of co-ordinates DC, y, z, the centre of the 

 base of the prism, and as rectangular axes, the axis of the prism 

 itself, and the two perpendiculars on the sides : the permanent 

 temperature v of the point m, whose co-ordinates are #, y, z, is 

 a function of three variables F (x, y, z) : it has by hypothesis a 

 constant value, when we suppose x nothing, whatever be the values 

 of y and z. Suppose we take for the unit of heat that quantity 

 which in the unit of time would emerge from an area equal to a 

 unit of surface, if the heated mass which that area bounds, and 

 which is formed of the same substance as the prism, were continu 

 ally maintained at the temperature of boiling water, and immersed 

 in atmospheric air maintained at the temperature of melting ice. 



We see that the quantity of heat which, in the permanent 

 state of the rectangular prism, flows, during a unit of time, across 

 a certain section perpendicular to the axis, has a determinate ratio 



