Page:Philosophical magazine 23 series 4.djvu/34

18 Prof. Maxwell on the Theory Molecular Vortices

whose density is $\rho$, and having its normal inclined to the axes of $z$; then the tangential force upon it will be

$\rho R\delta S\sin \theta =2T\delta S,\,$

(99)

T beings as before, the tangential force on each side of the surface. Putting $\rho ={\tfrac {1}{2\pi }}$ as in equation (34)^[1], we find

$R=-2\pi ma(e+2f)\,$

(100)

The displacement of electricity due to the distortion of the sphere is

$\Sigma \delta S{\tfrac {1}{2}}\rho t\sin \theta \,$ taken over the whole surface;

(101)

and if $h$ is the electric displacement per unit of volume, we shall have

${\frac {4}{3}}\pi a^{3}h={\frac {2}{3}}a^{4}e,$

(102)

or

$h={\frac {1}{2\pi }}ae;$

(103)

so that

$R=4\pi ^{2}m{\frac {e+2f}{e}}h,$

(104)

or we may write

$R=-4\pi E^{2}h,$

(105)

provided we assume

$E^{2}=-\pi m{\frac {e+2f}{e}}$

(106)

Finding $e$ and $f$ from (87) and (90), we get

$E^{2}=\pi m{\frac {3}{1+{\frac {5}{3}}{\frac {m}{\mu }}}}.$

(107)

The ratio of $m$ to $\mu$ varies in different substances; but in a medium whose elasticity depends entirely upon forces acting between pairs of particles, this ratio is that of 6 to 5, and in this case

$E^{2}=\pi m.\,$

(108)

When the resistance to compression is infinitely greater than the resistance to distortion, as in a liquid rendered slightly elastic by gum or jelly,

$E^{2}=3\pi m.$

(109)

The value of $E^{2}$ must lie between these limits. It is probable that the substance of our cells is of the former kind, and that we must use the first value of $E^{2}$, which is that belonging to

1. Phil. Mag. April 1861.