FIRST SECTION.

Of the free movement of heat in an infinite line.

342. Here we consider the movement of heat in a solid homogeneous mass, all of whose dimensions are infinite. The solid is divided by planes infinitely near and perpendicular to a common axis; and it is first supposed that one part only of the solid has been heated, that, namely, which is enclosed between two parallel planes A and B whose distance is g; all other parts have the initial temperature 0; but any plane included between A and B has a given initial temperature, regarded as arbitrary, and common to every point of the plane; the temperature is different for different planes. The initial state of the mass being thus defined, it is required to determine by analysis all the succeeding states. The movement in question is simply linear, and in direction of the axis of the plane ; for it is evident that there can be no transfer of heat in any plane perpendicular to the axis, since the initial temperature at every point in the plane is the same.

Instead of the infinite solid we may suppose a prism of very small thickness, whose lateral surface is wholly impenetrable to heat. The movement is then considered only in the infinite line which is the common axis of all the sectional planes of the prism.