Page:Scientific Memoirs, Vol. 2 (1841).djvu/499

the atmosphere exercises a perceptible influence on a galvanic circuit, or not, we will therefore enter into this case at greater length.

Since, according to §12, the magnitude of the electric current is given by the equation

{nowrap

$\mathrm {S} =\chi \omega .{\frac {du}{dx}}$,

|}}we have only in each separate case to obtain the value of ${\frac {du}{dx}}$ from the equation found for the determination of the electroscopic force, and to place it in the one above. Thus, for a circuit which has assumed its permanent state, but upon which the surrounding atmosphere exercises no sensible influence, according to §22,

$u={\frac {1}{2}}a.{\frac {e^{\beta x}-e^{-\beta x}}{e^{\beta l}-e^{-\beta l}}}+{\frac {1}{2}}b{\frac {e^{\beta x}+e^{-\beta x}}{e^{\beta l}+e^{-\beta l}}}$,

where $a$ represents the tension at the place of excitation, and $b$ the sum of the electroscopic forces immediately adjacent on both sides of the place of excitation. We hence obtain

$\mathrm {S} =\chi \omega \beta \left({\frac {1}{2}}a{\frac {e^{\beta x}+e^{-\beta x}}{e^{\beta l}-e^{-\beta l}}}+{\frac {1}{2}}b{\frac {e^{\beta x}-e^{-\beta x}}{e^{\beta l}+e^{-\beta l}}}\right)$.

This expression gives the magnitude of the current at each place of the circuit; but the law, according to which the alteration of the current at various places of the circuit is effected, may be rendered more easily intelligible in the following manner. If, for instance, we differentiate the equation

$\mathrm {S} =\chi \omega {\frac {du}{dx}}$,

we obtain the equation

${\frac {d\mathrm {S} }{dx}}=\chi \omega {\frac {d^{2}u}{dx^{2}}}$;

and by multiplying both together,

$\mathbf {S} {\frac {d\mathbf {S} }{du}}=\chi ^{2}\omega ^{2}{\frac {d^{2}u}{dx^{2}}}$.

If we now substitute for ${\frac {d^{2}u}{dx^{2}}}$ its value $\beta ^{2}u$, as obtained from the equation $0={\frac {d^{2}u}{dx^{2}}}-\beta ^{2}u$, we have

$\mathbf {S} {\frac {d\mathrm {S} }{du}}=\chi ^{2}\omega ^{2}\beta ^{2}u$;