61 Mb maxwell, ON FARADAY'S LINES OF FORCE. 



Examples. 



I. Theory of Electrical Images. 



The method of Electrical Images, due to Prof. W. Thomson*, by which the theory of 



spherical conductors has been reduced to great geometrical simplicity, becomes even more 



simple when we see its connexion with the methods of this paper. We have seen that the 



pressure at any point in a uniform medium, due to a spherical shell (radius = a) giving out 



a^ . 

 fluid at the rate of '^irPa^ units in unit of time, is kP — outside the shell, and kPa inside it, 



r 



where r is the distance of the point from the centre of the shell. 



If there be two shells, one giving out fluid at a rate 4nrPa\ and the other absorbing at the 



rate 4nrP'a'^, then the expression for the pressure will be, outside the shells, 



a' , a'^ 



p = 4:TrP 47rP — ; , 



r r 



where r and r are the distances from the centres of the two shells. Equating this expression 

 to zero we have, as the surface of no pressure, that for which 



r P'a'-' 



r Pa'' 



Now the surface, for which the distances to two fixed points have a given ratio, is a sphere 

 of which the centre O is in the line joining the centres of the shells CC produced, so that 



co = cc ^ — 



Pa'^-Fa'Y 

 and its radius 



_^^ Pa'.P^a' 



Pa']'-P'a''Y' ' 



If at the centre of this sphere we place another source of the fluid, then the pressure due to 

 this source must be added to that due to the other two ; and since this additional pressure 

 depends only on the distance from the centre, it will be constant at the surface of the sphere, 

 where the pressure due to the two other sources is zero. 



We have now the means of arranging a system of sources within a given sphere, so that 

 when combined with a given system of sources outside the sphere, they shall produce a given 

 constant pressure at the surface of the sphere. 



" See a series of papers " On the Mathematical Theory of Electricity," in the Cambridge and Dublin Math. Jour., begin- 

 ning March, 1848. 



