Page:On Faraday's Lines of Force.pdf/17

On this supposition we can easily calculate the kind of alteration which the introduction of the internal medium will produce; for wherever a unit tube enters the surface we must conceive a source producing fluid at a rate ${\frac {k'-k}{k}}$ , and wherever a tube leaves it we must place a sink annihilating fluid at the rate ${\frac {k'-k}{k}}$ , then calculating pressures on the supposition that the resistance in both media is k, the same as in the external medium, we shall obtain the true distribution of pressures very approximately, and we may get a better result by repeating the process on the system of pressures thus obtained.

(27) If instead of an abrupt change from one coefficient of resistance to another we take a case in which the resistance varies continuously from point to point, we may treat the medium as if it were composed of thin shells each of which has uniform resistance. By properly assuming a distribution of sources over the surfaces of separation of the shells, we may treat the case as if the resistance were equal to unity throughout, as in (23). The sources will then be distributed continuously throughout the whole medium, and will be positive whenever the motion is from places of less to places of greater resistance, and negative when in the contrary direction.

(28) Hitherto we have supposed the resistance at a given point of the medium to be the same in whatever direction the motion of the fluid takes place; but we may conceive a case in which the resistance is different in different directions. In such cases the lines of motion will not in general be perpendicular to the surfaces of equal pressure. If a, b, c be the components of the velocity at any point, and $\alpha ,\beta ,\gamma$ the components of the resistance at the same point, these quantities will be connected by the following system of linear equations, which may be called “equations of conduction,” and will be referred to by that name.

$a=P_{1}\alpha +Q_{3}\beta +R_{2}\gamma ,$

$b=P_{2}\beta +Q_{1}\gamma +R_{3}\alpha ,$

$c=P_{3}\gamma +Q_{2}\alpha +R_{1}\beta .$

In these equations there are nine independent coefficients of conductivity. In order to simplify the equations, let us put

$Q_{1}+R_{1}=2S_{1},$	$Q_{1}-R_{1}=2lT,$
............&c.	............&c.