Page:On Faraday's Lines of Force.pdf/15

Rh 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 multiplied 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 $$k=1$$.

(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