Page:On Faraday's Lines of Force.pdf/14

168 of (14) we can determine the distribution of sources to which a given distribution of pressures is due.

(20) We have next to shew that if we conceive any imaginary surface as ﬁxed in space and intersecting the lines of motion of the ﬂuid, we may substitute for the ﬂuid on one side of this surface a distribution of sources upon the surface itself without altering in any way the motion of the ﬂuid on the other side of the surface.

For if we describe the system of unit tubes which deﬁnes the motion of the ﬂuid, and wherever a tube enters through the surface place a unit source, and wherever a tube goes out through the surface place a unit sink, and at the same time render the surface impermeable to the ﬂuid, the motion of the ﬂuid in the tubes will go on as before.

(21) If the system of pressures and the distribution of sources which produce them be known in a medium whose resistance is measured by k, then in order to produce the same system of pressures in a medium whose resistance is unity, the rate of production at each source must be multiplied by k. For the pressure at any point due to a given source varies as the rate of production and the resistance conjointly; therefore if the pressure be constant, the rate of production must vary inversely as the resistance.

(22) On the conditions to be fulfilled at a surface which separates two media whose coefficients of resistance are k and k' .

These are found from the consideration, that the quantity of ﬂuid which ﬂows out of the one medium at any point ﬂows into the other, and that the pressure varies continuously from one medium to the other. The velocity normal to the surface is the same in both media, and therefore the rate of diminution of pressure is proportional to the resistance. The direction of the tubes of motion and the surfaces of equal pressure will be altered after passing through the surface, and the law of this refraction will be, that it takes place in the plane passing through the direction of incidence and the normal to the surface, and that the tangent of the angle of incidence is to the tangent of the angle of refraction as k'  is to k.

(23) Let the space within a given closed surface be ﬁlled with a medium different from that exterior to it, and let the pressures at any point of this compound system due to a given distribution of sources within and without