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 are 
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 9 and ¢ 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 O 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, 6. The area of the transverse sec- 
tion of the bar will be ad; and therefore if Y denote the strength, and 7 the inten- 
sity of the current in it, we have,— 
lag 
ab 
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 7 cos w, and 
2 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 CD, 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 toO 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. 27 
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