DYNAMICAL THEORY OF HEAT. 159 



blishes Proposition II., enunciated above in § 149, and shows the amount of the 

 electro-motive force producing the stated effect, in terms of T and T, the tempera- 

 tures of the two sides of the bar, the obliquity of the bar to the crystalline axis of 

 symmetry, and the thermo-electric properties of the substance ; since, if 6 and 

 (j> denote its thermo-electric powers, along the axis of symmetry, and along lines 

 perpendicular to this axis, at the temperature t, and w the inclination of this axis 

 to the length of the bar when the substance is at the temperature t, we have 



a = j ((f) — 6 ) sin o» cos w ... . (28). 



155. By an investigation exactly similar to that of § 115 which had reference to 

 non-crystalline linear conductors, we deduce the following expression for the 

 electro-motive force, when the ends of the bar are kept at temperatures T, T", from 

 the terminal thermal agency n, of a current investigated in § 153. 



J r t 



dt (29), 



where 



n = i (6 cos 2 ea + <p sin 2 «).... (30). 



§§ 156-170. On the Thermal Effects and the Thermo-electric Excitation of Electrical 

 Currents in Homogeneous Crystalline Solids. 



156. The Propositions I. and II., investigated above, suggest the kind of assump- 

 tions to be made regarding the reversible thermal effects of currents in uniformly 

 heated crystalline solids, and the electro-motive forces induced by any thermal 

 circumstances which cause inequalities of temperature in different parts. The 

 formulae expressing these agencies in the particular case which we have now in- 

 vestigated, guide us to the precise forms required to express those assumptions 

 in the most general possible manner. 



157. Let us first suppose a rectangular parallelepiped (a, b, c) of homogeneous 

 crystalline conducting matter, completely surrounded by continuous metal of the 

 standard thermo-electric quality touching it on all sides, to be traversed in any 

 direction by a uniform electric current, of which the intensity components parallel 

 to the three edges of the parallelepiped are h, i, j, and to be kept in all points at a 

 uniform temperature t. Then taking cp, 6, -v},, to denote the thermo-electric powers 

 of bars of the substance cut from directions parallel to the edges of the parallele- 

 piped, quantities which would be equal to one another in whatever directions 

 those edges are if the substance were non-crystalline ; and 6' 6", cp', <p", y, ^", 

 other elements depending on the nature of the substance with reference to the 

 directions of the sides of the parallelepiped, to which the name of thermo-electric 

 obliquities may be given, and which must vanish for every system of rectangular 



VOL. XXI. PART I. 2 U 



