170 ON FARADAY'S LINES OF FORCE. 



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 pres- 

 sure 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 zero 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. 



