Prof. Thomson on the Dijnamical Theory of Heat. 443 



which is essentially, in its magnetic chai-acter, dipolar, as thermo- 

 electrically dipolar also. 



§§ 172-181. On the general equations of Thermo-electric Action 

 in any homogeneous or heterogeneous crystallized or non-crystal- 

 lized solid. 



172. Let t denote the absolute temperatui-e at any point, 

 X, y, z, of a solid. Let d, (f>, f, &, <^', -f', &\ </>", -f" be the 

 values of the nine thermo-electric coefficients for the substance 

 at this point, quantities which may vary from point to point, 

 either by heterogeneousness of the solid, or in virtue of non- 

 uniformity of its temperature. Let h, i, j be the components of 

 the intensity of electric current through the same point [x, y, z). 



173. Then, applying equations (31) of § 157 to infinitely 

 small, contiguous, rectangular parallelopipeds in the neighbour- 

 hood of the point {x, y, z), and denoting by H dx dy ds the 

 resultant reversible absorption of heat occasioned by the electric 

 current across the infinitely small element dx dy dz, we find 



174. By the analysis of discontinuous functions, this expres- 

 sion may be applied not only to homogeneous or to continuously 

 varying heterogeneous substances, but to abrupt transitions from 

 one kind of substance to another. Still it may be convenient to 

 have formulae immediately applicable to such cases, and therefore 

 I add the following expression for the reversible thermal efi'ect 

 in any part of the bounding surface separating the given solid 

 from a solid of the standard metal in contact with it : — 



Q,=j{p{he + i<\>"-{-jf')+q{he' + i<^+jy) +r{h&' + i<\^' +jy^)\. (44), 



where Q denotes the quantity of heat absorbed per second per 

 unit of surfece at a point of the bounding surface, and {p, q, r) 

 the direction cosines of a normal to the surface at the same point. 



175. Equations (34) give explicitly the intrinsic electromotive 

 force at any point of the solid when the distribution of tempera- 

 ture is given ; but we must take into account also the reaction 

 proceeding from the surrounding matter, to get the efficient 

 electromotive force determining the current through any part of 

 the body. This reaction will be the electrostatical resultant 

 force due to accumulations of electricity at the bounding surface 

 and in the interior of the conducting mass throughout which the 



small in comparison with the amount by which the thermo-electric power 

 in the direction of magnetization diflfers from the thermo-electric power of 

 the same metal not magnetized. 



2G2 



