Page:Scientific Memoirs, Vol. 2 (1841).djvu/485

point of excitation, is known, and is equal to $b$, which case always occurs when the electroscopic force of the circuit is given at any one of its places, we obtain

$b={\frac {a\left(e^{\beta l}+e^{-\beta l}\right)}{e^{\beta l}-e^{-\beta l}}}+2d\left(e^{\beta l}+e^{-\beta l}\right);$

and after substitution and proper reduction,

$u={\frac {{\frac {1}{2}}a\left(e^{\beta x}-e^{-\beta x}\right)}{e^{\beta l}-e^{-\beta l}}}+{\frac {{\frac {1}{2}}b\left(e^{\beta x}+e^{-\beta x}\right)}{e^{\beta l}+e^{-\beta l}}},$

which for $b=0$, i. e. for a circuit left entirely to itself, changes into

$u={\frac {{\frac {1}{2}}a\left(e^{\beta x}-e^{-\beta x}\right)}{e^{\beta l}-e^{-\beta l}}}.$

These equations, which hold for a circuit homogeneous and prismatic in its whole extent, change when $\beta =0$ again into the above, where the influence of the atmosphere on the circuit was, under the circumstances given above, left out of consideration. Since $\beta ^{2}={\frac {b}{\chi }}\centerdot {\frac {c}{\omega }}$, it follows that the influence of the atmosphere on the galvanic circuit must be less, the smaller the conducting power of the atmosphere is in comparison to that of the circuit, and the smaller the quotient ${\frac {c}{\omega }}$ is. But the quotient ${\frac {c}{\omega }}$ expresses the relation of the surface of a disc of the conductor surrounded by the atmosphere to the volume of the same disc, and it might therefore appear that ${\frac {c}{\omega }}$ must constantly be infinitely small. However, it must not be forgotten that we have not here to deal with mathematical, but with physical determinations; for, strictly taken, $c$ does not represent a surface, but that portion of a disc of the circuit on which the atmosphere has direct influence, and $\omega$ in fact signifies nothing more than that part of a disc of the circuit which is traversed by the electricity continually passing through the circuit. In general, therefore, $c$ is indeed incomparably smaller than $\omega$; but where the electric current can only move forwards with the greatest difficulty, and on that account but very slowly, as is more or less the case in dry piles, the magnitude $c$ may, in accordance with what was stated in the preceding paragraph, become very nearly equal to $\omega$; for undoubtedly a gradual transition,