Prof. Thomson on the Dynamical Theory of Heat. 439 



ively. Then if we take 



ql=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 expressions for the component 

 electi'omotive forces of the parallelopiped 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 referencCj 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 = M^ +ve' +we"-^ 



-Y=u(f>" + v<f) +W(f)' I .... (34). 



— G = u-yfr' + vyfr" + iv^lr 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 electromotive force at any point in the substance, whether 

 the portion of it we are considering be limited and spherical, or 

 rectangular, or of anyother shape, or be continued to anyindefinite 

 extent by homogeneous or heterogeneous solid conducting matter 

 with any distribution of temperature through it. The compo- 

 nent electromotive force P along a diameter of the sphere in- 

 clined to the rectangular lines of reference at angles whose 

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



P = E^+Fm + G« (35), 



which may also be employed to transform the general expressions 

 for the components of the electromotive 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 phsenomena of a solid rotating about a fixed 

 point, and for electrostatical and for magnetic forces, in natural 

 crystals or in substances structurally ci'ystalline as regards elec- 

 tric or magnetic induction ? The following transformation, sug- 

 gested by Mr. Stokes's paper on the Conduction of Heat in 

 Crystals*, in which a perfectly analogous transformation is ap- 

 plied to the most general conceivable equations expressing flux 

 of heat in terms of variations of temperature along rectangular 

 lines of reference in a solid, will show the nature of the answer. 



164. The direction cosines of the line of greatest thermal 

 variation, or the perpendicular to the isothermal planes, are 



* Cambridge and Dublin Mathematical Journal, Nov. 1851. 



