Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/272

Magnetic Pair.

(3) Quantity of free magnetism, or strength of a pole . $m$

(4) Magnetic potential . . . . . . . $\Omega$

Electrokinetic Pair.

(5) Electrokinetic momentum of a circuit . . . $p$

(6) Electric current . . . . . . . $C$

Second Three Pairs.

Electrostatic Pair.

(7) Electric displacement (measured by surface-density) . ${\mathfrak {D}}$

(8) Electromotive force at a point . . . . ${\mathfrak {E}}$

Magnetic Pair.

(9) Magnetic induction . . . . . . ${\mathfrak {B}}$

(10) Magnetic force . . . . . . . ${\mathfrak {H}}$

Electrokinetic Pair.

(11) Intensity of electric current at a point . . . ${\mathfrak {C}}$

(12) Vector potential of electric currents . . . ${\mathfrak {A}}$

622.] The following relations exist between these quantities. In the first place, since the dimensions of energy are $\left[{\frac {L^{2}M}{T^{2}}}\right]$, and those of energy referred to unit of volume $\left[{\frac {M}{LT^{2}}}\right]$, we have the following equations of dimensions:

$[e\,E]=[mz,\Omega ]=[p\,C]=\left[{\frac {L^{2}M}{T^{2}}}\right],$

(1)

$[{\mathfrak {D}}\,{\mathfrak {E}}]=[{\mathfrak {B}}\,{\mathfrak {H}}]=[{\mathfrak {C}}\,{\mathfrak {A}}]=\left[{\frac {M}{LT^{2}}}\right].$

(2)

Secondly, since e, p and ${\mathfrak {A}}$ are the time-integrals of C, E, and ${\mathfrak {E}}$ respectively

$\left[{\frac {e}{C}}\right]=\left[{\frac {p}{E}}\right]=\left[{\frac {\mathfrak {A}}{\mathfrak {E}}}\right]=[T].$

(3)

Thirdly, since E, Ω, and p are the line-integrals of ${\mathfrak {E}}$, ${\mathfrak {H}}$, and ${\mathfrak {A}}$ respectively,

$\left[{\frac {e}{\mathfrak {E}}}\right]=\left[{\frac {\Omega }{\mathfrak {H}}}\right]=\left[{\frac {p}{\mathfrak {A}}}\right]=[L].$

(4)

Finally, since e, C, and m are the surface-integrals of ${\mathfrak {D}}$, ${\mathfrak {G}}$, and ${\mathfrak {B}}$ respectively,

$\left[{\frac {e}{\mathfrak {D}}}\right]=\left[{\frac {C}{\mathfrak {E}}}\right]=\left[{\frac {m}{\mathfrak {B}}}\right]=\left[L^{2}\right].$

(5)