Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/276

the same as Fourier's equation for the linear propagation of heat: so that the known solutions of Fourier's theory may be used in a new interpretation. If we substitute

$V=e^{2nt{\sqrt {-1}}+\lambda x}$,

we obtain

$\lambda =\pm (1+{\sqrt {-1}})(nCR)^{\frac {1}{2}}$;

and therefore a typical elementary solution of the equation is

$V=e^{-(nCR)^{\frac {1}{2}}x}\sin\{2nt-(nCR)^{\frac {1}{2}}x\}$.

The form of this solution shows that if a regular harmonic variation of potential is applied at one end of a cable, the phase is propagated with a velocity which is proportional to the square root of the frequency of the oscillations: since therefore the different harmonics are propagated with different velocities, it is evident that no definite "velocity of transmission" is to be expected for ordinary signals. If a potential is suddenly applied at one end of the cable, a certain time elapses before the current at the other end attains a definite percentage of its maximum value; but it may easily be shown^[1] that this retardation is proportional to the square of the length of the cable, so that the apparent velocity of propagation would be less, the greater the length of cable used.

The case of a telegraph line insulated in the air on poles is different from that of a cable; for here the capacity is small, and it is necessary to take into account the inductance. If in the general equation of telegraphy we write

$V=e^{nx{\sqrt {-1}}+\mu t}$,

we obtain the equation

$\mu =-{\frac {R}{2L}}\pm \left({\frac {R^{2}}{4L^{2}}}-{\frac {n^{2}}{CL}}\right)^{\frac {1}{2}}$;

as the capacity is small, we may replace the quantity under the radical by its second term: and thus we see that a typical elementary solution of the equation is

$V=e^{-{\frac {Rt}{2L}}}\sin {n\{x-(CL)^{-{\frac {1}{2}}}t\}}$;

1. This result, indeed, follows at once from the theory of dimensions.