224 ON THE CONDUCTION OP HEAT IN CRYSTALS.
points in which the lines, all equal and parallel to EE' , which connect the pairs of elements, cut the plane S, or as the cosine of the angle between the normal to S and the direction EE'. Let EE' = s] let f, m', ri, be the cosines of the angles which this line makes with the axes of #, y, z, and suppose S to be perpendicular to each of these axes in succession. We shall thus have for the partial fluxes fx,fy,fz, quantities proportional to Vdujds^ m'du/ds, n' du/ds, or to
o du j, , du Jt , du lf , du ,2 du f f du
7/o j, , Jt , lf , ,2 f f
I ~r~ + IM T- + 1 n "T~ > lm T~ + m TJ- + mn T > dx dy dz dx ay dz
7, fdu , ,du ,,2du In ~=~ + m n -7- + w -y . a« ai/ <t£
Hence, the coefficient of dujdy in the expression for the partial flux /^ is equal to the coefficient of dufdx in the expression for the partial flux /y, and the same applies to y, z, and to z9 x. This being true for each partial flux, will be true likewise for the total flux, and therefore the general expressions for the flux in three rectangular directions, with nine arbitrary constants, will, be reduced to the form (8), or the general expressions (6), referred to the thermic axes, to the form (7).
19. Let us further examine some of the consequences which would follow from the supposition that the expressions for the flux referred to the thermic axes have the general form (6). Conceive a crystalline mass, regarded as infinite, to be heated at one point according to any law, and let the source of heat be taken for origin. We have seen already that the succession of temperatures takes place in an infinite solid in exactly the same manner whether the expressions for the flux have the general form (6), or the more restricted form (7), and consequently, in the case supposed above, the temperature at a given time is some function of
If #, y, z, be the coordinates of any point in a line of motion, or line traced at a given instant from point to point in the direction of the flow of heat, doc, dy, dz, will be proportional to fx, fv, fz, which are given by (6), and in the present case du/dat, du/dy, du/dz,