Page:On Faraday's Lines of Force.pdf/20

174 the pressure is therefore S'p' . The whole work done by the ﬂuid may therefore be expressed by

$ W=\Sigma Sp-\Sigma S'p'$,

or more concisely, considering sinks as negative sources,

$ W=\Sigma(Sp)$,

(31) Let S represent the rate of production of a source in any medium, and let p be the pressure at any given point due to that source. Then if we superpose on this another equal source, every pressure will be doubled, and thus by successive superposition we ﬁnd that a source nS would produce a pressure np, or more generally the pressure at any point due to a given source varies as the rate of production of the source. This may be expressed by the equation

$p=RS $,

where R is a coefﬁcient depending on the nature of the medium and on the positions of the source and the given point. In a uniform medium whose resistance is measured by k,

$p=\frac{kS}{4\pi r},\; \therefore \; R=\frac{k}{4\pi r} $,

R may be called the coefficient of resistance of the medium between the source and the given point. By combining any number of sources we have generally

$p=\Sigma (RS) $. (32) In a uniform medium the pressure due to a source S

$p=\frac{k}{4\pi}\frac{S}{r} $.

At another source S'  at a distance r we shall have

$S'p=\frac{k}{4\pi}\frac{SS'}{r} =Sp' $,

if p’  be the pressure at S due to S' . If therefore there be two systems of sources $$\Sigma(S) \; and\; \Sigma(S')$$, and if the pressures due to the ﬁrst be p and to the second p' , then

$\Sigma(S'p)=\Sigma(Sp') $.

For every term S'p has a term Sp'  equal to it.