Page:On Faraday's Lines of Force.pdf/19

It is evident that all that has been proved in (14), (15), (16), (17), with respect to the superposition of different distributions of pressure, and there being only one distribution of pressures corresponding to a given distribution of sources, will be true also in the case in which the resistance varies from point to point, and the resistance at the same point is different in different directions. For if we examine the proof we shall find it applicable to such cases as well as to that of a uniform medium.

(29) We now are prepared to prove certain general propositions which are true in the most general case of a medium whose resistance is different in difl'erent directions and varies from point to point.

We may by the method of (28), when the distribution of pressures is known, construct the surfaces of equal pressure, the tubes of fluid motion, and the sources and sinks. It is evident that since in each cell into which a unit tube is divided by the surfaces of equal pressure unity of fluid passes from pressure $p$ to pressure $(p-1$) in unit of time, unity of work is done by the fluid in each cell in overcoming resistance.

The number of cells in each unit tube is determined by the number of surfaces of equal pressure through which it passes. If the pressure at the beginning of the tube be $p$ and at the end $p'$, then the number of cells in it will be $p-p'$. Now if the tube had extended from the source to a place where the pressure is zero, the number of cells would have been $p$, and if the tube had come from the sink to zero, the number would have been $p'$, and the true number is the difference of these.

Therefore if we find the pressure at a source $S$ from which $S$ tubes proceed to be $p$, $Sp$ is the number of cells due to the source $S$; but if $S'$ of the tubes terminate in a sink at a pressure $p'$, then we must cut off $S'p'$ cells from the number previously obtained. Now if we denote the source of $S$ tubes by $S$, the sink of $S'$ tubes may be written $-S'$, sinks always being reckoned negative, and the general expression for the number of cells in the system will be $\Sigma (Sp)$.

(30) The same conclusion may he arrived at by observing that unity of work is done on each cell. Now in each source $S,S$ units of fluid are expelled against a pressure $p$, so that the work done by the fluid in overcoming resistance is $Sp$. At each sink in which $S'$ tubes terminate, $S'$ units of fluid sink into nothing under pressure $p'$; the work done upon the fluid by