Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/345

formed of a single metal no current will be formed in it, however the section of the conductor and the temperature may vary in different parts.

Since in this case there is conduction of heat and consequent dissipation of energy, we cannot, as in the former case, consider this result as self-evident. The electromotive force, for instance, between two portions of a circuit might have depended on whether the current was passing from a thick portion of the conductor to a thin one, or the reverse, as well as on its passing rapidly or slowly from a hot portion to a cold one, or the reverse, and this would have made a current possible in an unequally heated circuit of one metal.

Hence, by the same reasoning as in the case of Peltier's phenomenon, we find that if the passage of a current through a conductor of one metal produces any thermal effect which is reversed when the current is reversed, this can only take place when the current flows from places of high to places of low temperature, or the math>reverse, and if the heat generated in a conductor of one metal in flowing from a place where the temperature is $x$ to a place where it is $y,$ is $H,$ then

$JH=RC^{2}t-S_{xy}Ct,$

and the electromotive force tending to maintain the current will be $S_{xy}$.

If $x,y,z$ be the temperatures at three points of a homogeneous circuit, we must have

$S_{yz}+S_{zx}+S_{xy}=0,$

according to the result of Magnus. Hence, if we suppose $z$ to be the zero temperature, and if we put

$Q_{x}=S_{xz},\quad \quad$ and $\quad \quad Q_{y}=S_{yz},$

we find

$S_{xy}=Q_{x}-Q_{y},$

where $Q_{x}$ is a function of the temperature $x$, the form of the function depending on the nature of the metal.

If we now consider a circuit of two metals $a$ and $b$ in which the temperature is $x$ where the current passes from $a$ to $b$, and $y$ where it passes from $b$ to $a$, the electromotive force will be

$F=P_{ax}-P_{bx}+Q_{bx}-Q_{by}+P_{by}-P_{ay}+Q_{ay}-Q{az},$

where $P_{ax}$ signifies the value of $P$ for the metal $a$ at the temperature $x$, or

$F=P_{ax}-Q_{ax}-(P_{ay}-Q_{ay})-(P_{bx}-Q_{bx})+P_{by}-Q_{by}.$

Since in unequally heated circuits of different metals there are in