ELECTBOSTATICS AND MAGNETISM. 303 



Poisson had already, by means of Laplace's powerful method of spherical 

 harmonics, determined, in the form of an infinite series, the distribution of 

 electricity on a sphere acted on by an electrified system. No one, however, 

 seems to have observed that when the external electrified system is reduced 

 to a point, the resultant external action is equivalent to that of this point, 

 together with an imaginary electrified point within the sphere, which Thomson 

 calls the electric image of the external point. 



Now if in an infinite conducting solid heat is flowing outwards uniformly 

 from a very small spherical source, and part of this heat is absorbed at another 

 small spherical surface, which we may call a sink, while the rest flows out in 

 all directions through the infinite solid, it is easy, by Fourier's methods, to 

 calculate the stationary temperature at any point in the solid, and to draw 

 the isothermal surfaces. One of these surfaces is a sphere, and if, in the 

 electrical problem, this sphere becomes a conducting surface in connection with 

 the earth, and the external source of heat is transformed into an electrified 

 point, the sink will become the image of that point, and the temperature and 

 flow of heat at any point outside the sphere will become the electric potential 

 and resultant force. 



Thus Thomson obtained the rigorous solution of electrical problems relating 

 to spheres by the introduction of an imaginaiy electrified system within the 

 sphere. But this imaginary system itself next became the subject of exami- 

 nation, as the result of the transformation of the external electrified system by 

 reciprocal radii vectores. By this method, called that of electrical inversion, the 

 solution of many new problems was obtained by the transformation of problems 

 already solved. A beautiful example of this method is suggested by Thomson 

 in a letter to M. Liouville, dated October 8, 1845, and published in Liouville's 

 Journal, for 1845, but which does not seem to have been taken up by any 

 mathematician, till Thomson himself, in a hitherto unpublished paper (No. xv. 

 of the book before us), wrote out the investigation complete. This, the most 

 remarkable problem of electrostatics hitherto solved, relates to the distribution 

 of electricity on a segment of spherical surface, or a bowl, as Thomson calls 

 it, under the influence of any electrical forces. The solution includes a very 

 important case of a flat circular dish, and of an infinite flat screen with a 

 circular hole cut out of it. 



If, however, the mathematicians were slow in making use of the physical 

 method of electric inversion, they were more ready to appropriate the geometrical 



