214 ON THE CONDUCTION OF HEAT IN CRYSTALS.
It follows from these equations, that if we suppose not only the temperatures at corresponding points of the two solids to be always the same, but also equal quantities of heat to flow in equal times and in corresponding directions across corresponding surfaces, the flow of heat in the auxiliary solid is what would naturally belong to an ordinary medium having an interior conductivity K.
The density of the auxiliary solid being disposable, we may take it to be the same as that of the given solid, in which case corresponding spaces will contain equal quantities of matter. It is only necessary further to suppose the auxiliary solid to be an ordinary medium having a specific heat c, and an interior conductivity K, in order that the motion of heat in the interior of the two solids should precisely correspond.
11. It remains only to investigate the condition which must be satisfied relatively to the surface of the auxiliary solid, in order that the two solids should perfectly correspond in every respect. Retaining the notation of Art. 9, let do- be the element of surface which corresponds to dS\ X, /u,, v, the direction-cosines corresponding to I, m, n. The quantity of heat which escapes across dS during the time dt is ultimately equal to h(u — v)dSdt, and this must be equal to the quantity which escapes across dcr. Hence it is sufficient to attribute to the auxiliary solid an exterior conductivity k, such that
*-&*
But we have
X fji V 1
Also IdS, \dcr, are the projections of dS, dcr, on the plane of yz, and these projections are proportional to *J(BG), K, or to *//C */A, whence we get from (15)
'
The first or second of these expressions will be employed according as we suppose I, m> n, or X, /^ v, given.