Prof. Thomson on the Dynamical Theory of Heat. 437 



160. To prove this, let us first consider the condition of a bar 

 of any of the three systems, taken alone, and put in the same 

 thermal circumstances as those in which each bar of the same 

 system exists in the compound mass. If, for instance, we take 

 a bar of the first system, we must suppose the temperature to 

 vary at the rate u per unit of space along its length ; at the rate 

 V across it, perpendicularly to two of its sides ; and at the rate w 

 across it, perpendicularly to its other two sides. If / be its 

 length, and e the breadth of each side, its ends will dift'er in 

 temperature by ul; corresponding points in one pair of its sides 

 by ve, and corresponding points in the other pair of sides by we. 

 Now it is easily proved that the longitudinal electromotive force 

 (that is, according to the definition, the electromotive force be- 

 tween conductors of the standard metal connected with its ends) 

 would, with no difi'erence of temperatures between its sides, and 

 the actual difference ul between its ends, be equal to |(ct| + ot/jm/, 

 if only the first of the zigzag conductors existed imbedded in the 

 bar, or equal to ^(icti" + ot, "')«/, if only the second ; and since the 

 two have equal resistances to conduction, and are connected by 

 a little square disc of the standard metal, it follows that the 

 longitudinal electromotive force of the actual bar, with only the 

 longitudinal variation of temperature, is 



Again, with only the lateral variation ve, we have in one of the 

 zigzags a little thermo-electric battery, of a number of elements 



amounting to the greatest integer in ^, which is sensibly equal 

 to ^, since the value of this is infinitely great ; the electromo- 

 tive force of each element is (otj — ■Br/jre ; and therefore the whole 

 electromotive force of the zigzag is 



— X (OT-OTi>e, or iZx (ct,-ct,>. 



This battery is part of a complete circuit with the little terminal 

 squares and the other zigzag, and therefore its electromotive 

 force will sustain a current in one direction through itself, and 

 in the contrary through the second zigzag ; but since the resist- 

 ances are equal in the two zigzags, and those of the terminal 

 connexions may be neglected, just half the electromotive force 

 of the first zigzag, being equal to the action and reaction between 

 the two parts of the circuit, must remain ready to act between 

 conductors applied to the terminal discs of the standard metal. 

 In the circumstances now supposed, the second zigzag is through- 

 out at one tempcraturCj and therefore has no intrinsic electro- 



