Page:On Faraday's Lines of Force.pdf/16

170 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 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.