It was **mathematically** treated by Louis Carre in 1705 and Koersma in 1741.

He also made important contributions to the mathematical theory of electrodynamics, and in papers published in 1845 and 1847 established **mathematically** the laws of the induction of electric currents.

Oliver Heaviside showed **mathematically** that uniformly-distributed inductance in a telephone line would diminish both attenuation and distortion, and that if the inductance were great enough and the insulation resistance not too high the circuit would be distortionless, while currents of all frequencies would be equally attenuated.

Amongst the earliest mechanical contrivances of Fraunhofer was a machine for polishing **mathematically** uniform spherical surfaces.

This equation, which is **mathematically** deducible from the kinetic theory of gases, expresses the behaviour of gases, the phenomena of the critical state, and the behaviour of liquids; solids are not accounted for.