168 PROFESSOR W. THOMSON ON THE 



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 tem- 

 perature. 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 parallelepipeds in the neighbourhood of the point (x, y, z), and de- 

 noting by II dx dy dz the resultant reversible absorption of heat occasioned by 

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



H 



= j{£( h6 + i ^' + J^ + ^^ & + i ^ + J^ + Tz^ 6 '' + i ^ + ^} • • ( - 43 '- 



174. By the analysis of discontinuous functions this expression 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 conve- 

 nient to have formulae immediately applicable to such cases, and therefore I add 

 the following expression for the reversible thermal effect in any part of the 

 bounding surface separating the given solid from a solid of the standard metal in 

 contact with it. 



Q=i{p(hd + i( P" + J V) + q(h6' + i(p+jy') + r(h6" + i(l)' + j^) | . . (44), 



where Q denotes the quantity of heat absorbed per second per unit of surface at 

 a point of the bounding surface, and {p, q, r) the direction cosines of a normal at 

 the point. 



175. Equations (34) give explicitly the intrinsic electro-motive force at any 

 point of the solid, when the distribution of temperature is given ; but we must 

 take into account also the reaction proceeding from the surrounding matter, to 

 get the efficient electro-motive force determining the current through any part of 

 the body. This reaction will be the electro-statical resultant force due to accu- 

 mulations of electricity at the bounding surface and in the interior of the con- 

 ducting mass throughout which the electrical circuits are completed. Hence if V 

 denote the electrical potential at (x, y, z) due to these accumulations, the compo- 

 nents of the reactional electro-motive force are — 



_ d V d V _ d V m 



dx ' dy ' dz ' 



and the components of the efficient electro-motive force in the solid, are therefore — 



E _ dV_ F _dV G _dJ_ 

 dx dy ' dz ' 



where E, F, G are given by the following equations, derived from (34) by substi- 

 tuting for u, 

 troduced : — 



dt dt dt 



tuting for zi, v, u; their values -r-, -j-, -j- , in terms of the notation now in 



