x
perhaps, the most important refers to double refraction in strained
uncrystalline bodies.
Neumann next turned his attention to electricity, and in two im- portant papers, published in 1845 and 1847, established the laws of induction of electrical currents. We meet here, for the first time, with the "electrodynamic potential." It is shown how currents, induced in one circuit either by the motion of conductors carrying electric currents, or by a change in the intensity of the current, may be deduced from one function depending on the relative position of the conductors, and that this function will also determine the mechanical forces acting between the conductors. To appreciate fully the great advance which was made by these two memoirs, it is necessary to realise that the papers were published before it had been shown, by Helmholtz and Lord Kelvin, how the principle of the con servation of energy may be utilised in the treatment of the problem. It may also be pointed out that Neumanu's investigations are deduced from Lenz' laws, which are direct consequences of the principle of energy; so that Neumann's treatment may, indirectly, be said to depend on that principle.
Neumann was the first to solve the problem of the magnetisation induced in an ellipsoid of revolution under the action of any mag- netic forces. Other important contributions relate to the functions known as spherical harmonics. It is a matter for regret that his first paper on that subject (Astronomische Nachrichten,' 1838) was completely overlooked by magneticians until Ad. Schmidt recently drew attention to it advantage have been employed in the treatment of terrestrial mag netism, may be explained by reference to the simpler problem of expanding a function of one variable by means of Fourier's series. For instance, if the daily changes of temperature are to be expressed in such a series from hourly readings of the thermometers, a very simple and well-known process leads to the determination of the constants Neumann's investigations led him to an analogous process for the expansion of a function in a series of spherical harmonics, the func tions having known values at the points of intersection of certain latitude and longitude circles on a sphere.f
Neumann's last publication was a memoir (edited by his son, C Neumann), 'Beiträge zur Theorie der Kugelfunctionen,' which con- tains many interesting theoretical researches on that subject.
The method which might with great Neumann's initials are often incorrectly given; thus, in the text of Maxwell's Electricity and Magnetism' (second edition) he is uniformly quoted as J. Neumann.
t In both the problems mentioned the values of the constante are really indeter- minate, but the solution gives, under certain assumptions, their most probable values. Care should be taken that in any actnal problem the assumptions are really justified
