ON FARADAY'S LINES OF FOKCE. 169 



the surface be given ; it is required to determine a distribution of sources which 

 would produce the same system of pressures in a medium whose coefficient of 

 resistance is unity. 



Construct the tubes of fluid motion, and wherever a unit tube enters either 

 medium place a unit source, and wherever it leaves it place a unit sink. Then 

 if we make the surface impermeable all will go on as before. 



Let the resistance of the exterior medium be measured by k, and that of 

 the interior by k'. Then if we multiply the rate of production of all the sources 

 in the exterior medium (including those in the surface), by k, and make the 

 coefficient of resistance unity, the pressures will remain as before, and the same 

 will be true of the interior medium if we multiply all the sources in it by k', 

 including those in the surface, and make its resistance unity. 



Since the pressures on both sides of the surface are now equal, we may 

 suppose it permeable if we please. / 



We have now the original system of pressures produced in a uniform medium 

 by a combination of three systems of sources. The first of these is the given 

 external system multiplied by k, the second is the given internal system multi- 

 plied by k', and the third is the system of sources and sinks on the surface 

 itself. In the original case every source in the external medium had an equal 

 sink in the internal medium on the other side of the surface, but now the 

 source is multiplied by k and the sink by k', so that the result is for every 

 external unit source on the surface, a source = (k k'). By means of these three 

 systems of sources the original system of pressures may be produced in a medium 

 for which k=l. 



(24) Let there be no resistance in the medium within the closed surface, 

 that is, let k' = 0, then the pressure within the closed surface is uniform and 

 equal to p, and the pressure at the surface itself is also p. If by assuming 

 any distribution of pairs of sources and sinks within the surface in addition to 

 the given external and internal sources, and by supposing the medium the same 

 within and without the surface, we can render the pressure at the surface uni- 

 form, the pressures so found for the external medium, together with the uniform 

 pressure p in the internal medium, will be the true and only distribution of 

 pressures which is possible. 



For if two such distributions could be found by taking different imaginary 

 distributions of pairs of sources and sinks within the medium, then by taking 

 VOL. I. 22 



