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 Neumann'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 ( 4 Astronomische Nachrichten,' 1838) was 

 completely overlooked by magneticians until Ad. Schmidt recently 

 drew attention to it. The method which might with great 

 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), ' Beitrage zur Theorie der Kugelfunctionen,' which con- 

 tains many interesting theoretical researches on that subject. 



* 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. 



f In both the problems mentioned the values of the constants are really indeter- 

 minate, but the solution gives, under certain assumptions, their most probable 

 values. Care should be taken that in any actual problem the assumptions are really 

 justified. 



