164 PROFESSOR W. THOMSON ON THE 



line solid ; let us now, instead of a parallelepiped imbedded in the standard metal, 

 consider an insulated sphere of the crystalline substance, with sources of heat 

 and cold applied at its surface, so as to maintain a uniform variation of temper- 

 ature in all lines perpendicular to the parallel isothermal planes. Let the rate of 

 variation of temperature per unit of length, perpendicular to the isothermal sur- 

 faces, be q, and let the cosines of the inclinations of this direction to the three rect- 

 angular directions in the substance to which the edges of the parallelepiped first 

 considered were parallel, and which we shall now call the lines of reference, be 

 I, m, n, respectively. Then if we take 



q I = u , qm = v , qn = w , 



the substance of the sphere will be in exactly the same thermal condition as an equal 

 spherical portion of the parallelepiped ; and it is clear that the preceding expres- 

 sions for the component electro-motive forces of the parallelepiped will give the 

 electro-motive forces of the sphere between the pairs of points at the extremities 

 of diameters coinciding with the rectangular lines of reference, if we take each of 

 the three quantities, a, b, c, equal to the diameter of the sphere. Calling this 

 unity, then we have 



— E = u 6 + v 6' + w 6" 



— F = u(f)" + v(p+w(f>' \ , . . (34), 



— G = u -4/ + v 4" + vo 4> j 



According to the definition given above (§ 144, Def. 3), it appears that these 

 quantities, E, F, G, are the three components of the intrinsic electro-motive force at 

 any point in the substance, whether the portion of it we are considering be limited 

 and spherical, or rectangular, or of any other shape, or be continued to any in- 

 definite extent by homogeneous or heterogeneous solid conducting matter with 

 any distribution of temperature through it. The component electro-motive force 

 P along a diameter of the sphere inclined to the rectangular lines of reference at 

 angles whose cosines are I, m, n, is of course given by the equation 



P = EZ + Fm + G» (35), 



which may also be employed to transform the general expressions for the compo- 

 nents of the electro-motive force to any other lines of reference. 



163. A question now naturally presents itself, Are there three principal axes at 

 right angles to one another in the substance possessing properties of symmetry, 

 with reference to the thermo-electric qualities, analogous to those which have been 

 established for the dynamical phenomena of a solid rotating about a fixed point, 

 and for electro-statical and for magnetic forces, in natural crystals or in sub- 

 stances structurally crystalline as regards electric or magnetic induction ? The 

 following transformation, suggested by Mr Stokes' paper on the Conduction of 

 Heat in Crystals,* in which a perfectly analogous transformation is applied to the 



* Cambridge and Dublin Mathematical Journal. 



