DYNAMICAL THEORY OF HEAT. 163 
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 a , which is sensibly equal to = , since the value of this is infinitely great ; the 
electro-motive force of each element is (a, — @,') ve; and, therefore, the whole 
electro-motive force of the zigzag is = x(@—-@,') ve, or $1 x (@,—@,)v. This 
battery is part of a complete circuit with the little terminal squares and the other 
zigzag, and therefore its electro-motive force will sustain a current in one direc- 
tion through itself, and in the contrary through the second zigzag; but since the 
resistances are equal in the two zigzags, and those of the terminal connections may 
be neglected, just half the electro-motive 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 throughout at one temper- 
ature, and therefore has no intrinsic electro-motive force; and the resultant in- 
trinsic electro-motive force of the bar is therefore 
41 (@,-— w,')v. 
Similarly, if there were only the lateral variation we of temperature in the bar, 
we should find a resultant longitudinal electro-motive force equal to 
41 (@,’-— @,”) wv. 
Tf all the three variations of temperature are maintained simultaneously, each 
will produce its own electro-motive force, as if the others did not exist, and the 
resultant electro-motive force due to them all will therefore be,— 
U 
7 {@, te BD,’ se @," ae @,”) u+ (@, —s @,') uv + (@," Ls w," w} ‘ 
This being the electro-motive force of each bar of the first system in any of 
the cubes composing the actual solid, must be the component electro-motive force 
of each cube in the direction to which they are parallel; and, therefore, 
at { (@, + B/ +O," + Out (@, — Wy) v + (@,"-B,”) wh 
must be the component electro-motive force of the entire parallelepiped in the same 
direction. Similar expressions give the component electro-motive forces parallel 
to the edges 6 and ¢ of the solid, which are similarly produced by the bars of the 
second and third systems, and we infer the proposition which was to be proved. 
161. Cor. By choosing metals of which the thermo-electric relations, both to the 
standard metal and to one another, vary, we may not only make the nine co- 
efficients have any arbitrarily given values for a particular temperature, but we 
may make them each vary to any extent with a given change of temperature. 
162. For the sake of convenience in comparing the actual phenomena of ther- 
mo-electric force in different directions presented by an unequally heated crystal- 
VOL. XXI. PART I. 2x 
