212             ON  THE CONDUCTION  OF  HEAT IN  CRYSTALS.
fore will vanish compared with dS> since S vanishes compared with the linear dimensions of dS. Hence, even if the quantity of heat which entered by the cylindrical surface varied as the surface, it would vanish in the limit compared with, the quantity which escapes by dS. In fact, however, even if we suppose 8 comparable with the linear dimensions of dS, it may be shewn that the total quantity of heat which enters by the cylindrical surface is of the order SdS, because ultimately the quantity of heat which enters across that portion of the cylindrical surface for which the flux is positive is equal to the quantity which escapes across the remainder. Lastly, the gain or loss of heat by the element during a given time varies ultimately as the volume of the element. Hence, ultimately, the quantity of heat which enters the element across the plane through P, during the time dt, is equal to the quantity which escapes across dS. The former will be ultimately equal to dSdt multiplied by the value of the flux obtained from the general formulae (1) and (7) by taking x, y, z to denote the coordinates of P', and I, m, n, the direction-cosines of P'N. We may assume the latter to be proportional to the difference u—v between the temperature u of the solid at the point P^r and the temperature v of the surrounding space, and may accordingly express it by h(u — v)dSdt. Hence we have for the required condition,
+h(u-v) = 0 ......... (10).
The quantity h denotes the exterior conductivity of the solid. It is a certain function of I, m, n, the form of which M. Duhamel did not attempt to investigate, nor am I going to attempt the investigation myself. If however the crystal be covered with a thin coating of some other substance, sufficient to stop all direct radiation from the crystal into the surrounding space, h will depend upon the nature of the coating. In either case h will be constant throughout any plane face by which the crystal may be bounded.
10. Let us return to the consideration of the propagation of heat in the interior of the mass. Imagine the coordinates 0, y, z, of any point altered in the ratios of *JA to »JK, *JB to