Page:Scientific Memoirs, Vol. 1 (1837).djvu/475

(III)″

$q=q_{0}+{\frac {4\pi }{3k}}\Sigma (g{\bar {\omega _{\nu }}}+fq_{\nu })\delta _{\nu }^{3}{\frac {e^{-\alpha r_{\nu }}}{r_{\nu }}}.$

where the sums $\Sigma$ are to be extended to all the molecules, including the first.

This last equation determines what the density of the æther must be at each point $x\,y\,z$, in order that it may be in equilibrium when it is submitted to the action of the spherical molecules of matter. The value of this density consists of different terms, each of which is due to a particular molecule, and represents its proper atmosphere. As the quantity of æther diffused through the immensity of space may be considered as infinite, the atmosphere formed by each molecule for itself is always the same, and its density is only superadded to that which the æther in the same places owes to other causes. According to the nature of the molecular actions, the value of the coefficient $\alpha ={\sqrt {\frac {4\pi f}{k}}}$ should be considered as very great: hence it follows that the density of each atmosphere will be incomparably greater when quite near or in contact with the molecule, and will decrease very rapidly as its distance from the molecule increases. This circumstance enables us to determine with ease, by approximation, the value of $q_{\nu }$, or the density of the æther at the surface of any molecule whatsoever, on the supposition that the molecules are not too near each other. If, for instance, we make $r=\delta$ in the term answering to the first molecule, and $r_{1}=\mathrm {r_{1}}$, $r_{2}=\mathrm {r_{2}}$ … $r_{\nu }=\mathrm {r_{\nu }}$ in the other terms, all these will be very small in comparison with the first, and by neglecting them we shall have very nearly

$\mathrm {q} =q_{0}+{\frac {4\pi }{3k}}(g{\bar {\omega }}+f\mathrm {q} )\delta ^{2}$

whence we derive

(6)

$\mathrm {q} ={\frac {q_{0}+{\frac {4\pi }{3k}}g{\bar {\omega }}\delta ^{2}}{1+{\frac {4\pi }{3k}}f\delta ^{2}}}.$

6. We are now in a condition to consider the equilibrium of any molecule whatever, such as it is given by the equations (II).

The quantity $\epsilon$ under the double integral in these equations must be replaced by ${\frac {1}{2}}kq^{2}$. Let us represent the coordinates $x$, $y$, $z$, so far as they belong to the points in contact with the surface of the molecule, by $\mathrm {x} +\xi$, $\mathrm {y} +\eta$, $\mathrm {z} +\zeta$; $\mathrm {x}$, $\mathrm {y}$, $\mathrm {z}$ being the coordinates of its centre: by developing the expression for $q$, and stopping, because of the smallness of the molecule, at the first terms, we shall be able to take

$q^{2}=\mathrm {q} ^{2}+2\mathrm {q} {\frac {d\mathrm {q} }{d\mathrm {x} }}\xi +2\mathrm {q} {\frac {d\mathrm {q} }{d\mathrm {y} }}\eta +2\mathrm {q} {\frac {d\mathrm {q} }{d\mathrm {z} }}\zeta .$