88 Mr maxwell, ON FARADAY'S LINES OF FORCE. 



(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 uniform, 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 the difference of the two 

 for a third distribution, we should have the pressure of the bounding surface constant in 

 the new system and as many sources as sinks within it, and therefore whatever fluid flows 

 in at any point of the surface, an equal quantity must flow out at some other point. 



In the external medium all the sources destroy one another, and we have an infinite 

 medium without sources surrounding the internal medium. The pressure at infinity is zero, 

 that at the surface is constant. If the pressure at the surface is positive, the motion of 

 the fluid must be outwards from every point of the surface ; if it be negative, it must flow 

 inwards towards the surface. But it has been shewn that neither of these cases is possible, 

 because if any fluid enters the surface an equal quantity must escape, and therefore the 

 pressure at the surface is zero in the third system. 



The pressure at all points in the boundary of the internal medium in the third case 

 is therefore zero, and there are no sources, and therefore the pressure is everywhere zero, 

 by (16). 



The pressure in the bounding surface of the internal medium is also zero, and there 

 is no resistance, therefore it is ^ero throughout ; but the pressure in the third case is the 

 difference of pressures in the two given cases, therefore these are equal, and there is only 

 one distribution of pressure which is possible, namely, that due to the imaginary distribution 

 of sources and sinks. 



(25) When the resistance is infinite in the internal medium, there can be no passage 

 of fluid through it or into it. The bounding surface may therefore be considered as 

 impermeable to the fluid, and the tubes of fluid motion will run along it without cutting it. 



If by assuming any arbitrary distribution of sources within the surface in addition to 

 the given sources in the outer medium, and by calculating the resulting pressures and 

 velocities as in the case of a uniform medium, we can fulfil the condition of there being 

 no velocity across the surface, the system of pressures in the outer medium will be the 

 true one. For since no fluid passes through the surface, the tubes in the interior are 

 independent of those outside, and may be taken away without altering the external motion. 



(26) If the extent of the internal medium be small, and if the difference of resistance 

 in the two media be also small, then the position of the unit tubes will not be much 

 altered from what it would be if the external medium filled the whole space. 



