Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/461

Formulae of Gauss and Weber.

851.] The first of these expressions, (18), was discovered by Gauss^[1] in July 1835, and interpreted by him as a fundamental law of electrical action, that 'Two elements of electricity in a state of relative motion attract or repel one another, but not in the same way as if they are in a state of relative rest.' This discovery was not, so far as I know, published in the lifetime of Gauss, so that the second expression, which was discovered independently by W.Weber, and published in the first part of his celebrated Elektrodynamische Maasbestimmungen^[2] , was the first result of the kind made known to the scientific world.

852.] The two expressions lead to precisely the same result when they are applied to the determination of the mechanical force between two electric currents, and this result is identical with that of Ampère. But when they are considered as expressions of the physical law of the action between two electrical particles, we are led to enquire whether they are consistent with other known facts of nature.

Both of these expressions involve the relative velocity of the particles. Now, in establishing by mathematical reasoning the well-known principle of the conservation of energy, it is generally assumed that the force acting between two particles is a function of the distance only, and it is commonly stated that if it is a function of anything else, such as the time, or the velocity of the particles, the proof would not hold.

Hence a law of electrical action, involving the velocity of the particles, has sometimes been supposed to be inconsistent with the principle of the conservation of energy.

853.] The formula of Gauss is inconsistent with this principle, and must therefore be abandoned, as it leads to the conclusion that energy might be indefinitely generated in a finite system by physical means. This objection does not apply to the formula of Weber, for he has shewn^[3] that if we assume as the potential energy of a system consisting of two electric particles,

$\psi ={\frac {ee'}{r}}\left[1-{\frac {1}{2c^{2}}}\left({\frac {{\mathfrak {d}}r}{{\mathfrak {d}}t}}\right)^{2}\right],$

(20)

the repulsion between them, which is found by differentiating this quantity with respect to r, and changing the sign, is that given by the formula (19).

1. Werke (Gottingen edition, 1867), vol.v. p.616.
2. Abh. Leibnizens Ges., Leipzig (1846).
3. Pogg. Ann., lxxiii. p. 229 (1848).

* Werke (Gottingen edition, 1867), vol.v. p.616. † Abh. Leibnizens Ges., Leipzig (1846). ‡ Pogg. Ann., lxxiii. p. 229 (1848).