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242

The Mathematical Electricians of the

energy is a universal principle of nature: that the kinetic and potential energy of dynamical systems may be converted into heat according to definite quantitative laws, as taught by Rumford, Joule, and Robert Mayer^[1]; and that any of these forms of energy may be converted into the chemical, electrostatic, voltaic, and magnetic forces. The latter Helmholtz examined systematically.

Consider first the energy of an electrostatic field. It will be convenient to suppose that the system has been formed by continually bringing from a very great distance infinitesimal quantities of electricity, proportional to the quantities already present at the various points of the system; so that the charge is always distributed proportionally to the final distribution. Let e typify the final charge at any point of space, and V the final potential at this point. Then at any stage of the process the charge and potential at this point will have the values λc and λV, where λ denotes a proper fraction. At this stage let charges edλ be brought from a great distance and added to the charges λe. The work required for this is

$\Sigma ed\lambda .\lambda V$ ,

so the total work required in order to bring the system from infinite dispersion to its final state is

$\Sigma eV.\int _{0}^{1}\lambda d\lambda$ , or ${\tfrac {1}{2}}\Sigma eV$ .

By reasoning similar to that used in the case of electrostatic distributions, it may be shown that the energy of a magnetic field, which is due to permanent magnets and which also contains bodies susceptible to magnetic induction, is

${\tfrac {1}{2}}\iiint \rho _{0}\phi \ dx\ dy\ dz$ ,

where ρ₀ denotes the density of Poisson's equivalent magnetiza-

↑ Julius Robert Mayer (b. 1814, d. 1878), who was a medical man in Heilbronn, asserted the equivalence of heat and work in 1842, Annal. d. Chemie, xlii, p. 233; his memoir, like that of Helmholtz, was first declined by the editors of the Annalen der Physik. An English translation of one of Mayer's memoirs was printed in Phil. Mag. xxv (1863), p. 493.

[1] Julius Robert Mayer (b. 1814, d. 1878), who was a medical man in Heilbronn, asserted the equivalence of heat and work in 1842, Annal. d. Chemie, xlii, p. 233; his memoir, like that of Helmholtz, was first declined by the editors of the Annalen der Physik. An English translation of one of Mayer's memoirs was printed in Phil. Mag. xxv (1863), p. 493.

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