DYNAMICAL THEORY OF HEAT. 



155 



Prop. II. If the two sides of such a bar be kept at different temperatures, and 

 a homogeneous conducting arc be applied to points of the ends which are at the 

 same temperature, a current will be produced along the bar, and through the arc 

 completing the circuit. 



150. For proving these propositions, it will be convenient to investigate fully 

 the thermo-electric agency experienced by a bar cut obliquely from a crystalline 

 substance possessing an axis of symmetry, when placed longitudinally in a circuit 

 of which the remainder is composed of the standard metal, and kept with either 

 its sides or its ends unequally heated. Let and <P denote the thermo-electric 

 powers of two bars cut from the given substance in directions parallel and per- 

 pendicular to its axis of symmetry respectively. Let us suppose the actual bar to 

 be of rectangular section with two of its opposite sides perpendicular to the plane 

 of its length, and the axis of symmetry of its substance. Let a longitudinal section 

 in this plane be represented by the accompanying diagram ; let A or any line 

 parallel to it be the direction of the axis of symmetry through any point ; and let 

 w denote the inclination of this line to the length of the bar. Let the breadth of 

 the two opposite sides of the bar perpendicular to the plane of the diagram be 

 denoted by a, and in the plane of the diagram, b. The area of the transverse sec- 

 tion of the bar will be a b ; and therefore if 7 denote the strength, and i the inten- 

 sity of the current in it, we have, — 



^ = 



7 



a b 



151. We may suppose the current, itself parallel to the length of the bar 

 and in the direction from left to right 

 of the diagram, to be resolved, at any 

 point P at the side of the bar, into 

 two components in directions parallel 

 and perpendicular to O A, of which 

 the intensities will be i cos w, and 

 i sin w, respectively. The former of 

 these components may be supposed to 

 belong to a system of currents crossing 

 the bar in lines parallel to OA and 

 passing out of it, across the side C D, into a conductor of the standard metal ; 

 and the latter, to a system of currents entering the bar across C D, from the same 

 conductor of standard metal, and crossing it in lines perpendicular to A. The 

 resultant current in the supposed standard metal beside the bar will clearly be 

 parallel to the length, and can therefore (this metal being non-crystalline) pro- 

 duce no effect influencing the thermal agency at the side of the bar or within it. 

 The inclinations of the currents to a perpendicular to the separating plane of the 

 two metals being respectively 90°- w and «, their strength per unit of area of this 



VOL. XXI. PART I. 2 T 



