Page:Scientific Memoirs, Vol. 1 (1837).djvu/472

of the formula (5) from $\theta '=0$ to $\theta '=\pi$, from $\psi '=0$ to $\psi '=2\pi$, and from $r=0$ to $r=\infty$.

Let us begin with performing the integrations of the formula (V). In consequence of the quantity $g{\bar {\omega }}+fq$ being considered as constant, and as the spherical form of the molecules renders $\rho$ independent of $\omega$ and $\phi$, all the terms of the second and third line of this formula will vanish, and it being observed that we always have

$\int _{0}^{\pi }\int _{0}^{2\pi }\Pi _{n}\sin \omega \,d\omega \,d\phi =0,$

unless in the case of $\Pi ^{0}=1$, which gives

$\int _{0}^{\pi }\int _{0}^{2\pi }\Pi _{0}\sin \omega \,d\omega \,d\phi =4\pi ;$

the expression for $G$ will become $G={\frac {4\pi }{3}}{\frac {(g{\bar {\omega }}+fq)\delta ^{3}}{r}}$, $\delta$ representing the semidiameter of the molecule.

This integral has been obtained under the supposition that the origin of the coordinates is in the centre of the molecule; but the origin may be transferred to any point whatever, by restoring, instead of $r$, its general expression, and writing

(V)'

$G={\frac {4\pi (g{\bar {\omega }}+fq)\delta ^{3}}{3\{(x-\mathrm {x} )^{2}+(y-\mathrm {y} )^{2}+(z-\mathrm {z} )^{2}\}^{\frac {1}{2}}}};$

where $\mathrm {x}$, $\mathrm {y}$, $\mathrm {z}$ represent the coordinates of the centre of the molecule.

Before we proceed to the expression for $F$, we had better clearly define the signification of the term $q$ which it contains. We must consider this quantity $(q)$ such as it is given by the equation (III), not as the entire value of the density of the æther, but as the value only of its excess or deficiency above or below the sensibly uniform density which the æther diffused in equilibrium is supposed to have in that part of space. If we represent the latter density by $q_{0}$, the equations (III) and (VI), while we suppress the terms due to the quantities $G$, $G_{1}$, $G_{2}$, &c., must be satisfied by the substitution of $q=q_{0}$: and that, in order that the æther may remain in equilibrium spontaneously, or in consequence of the action of the forces not expressed, whose centres must be supposed to be at a very great distance. If, therefore, we take the difference between the equations resulting from this substitution and the original equations themselves, we shall have

(III)'

$k(q-q_{0})=-F+G+G_{1}+G_{2}\ldots \ldots +G_{\nu }+\mathrm {\&c.}$

${\frac {d^{2}(q-q_{0})}{dx^{2}}}+{\frac {d^{2}(q-q_{0})}{dy^{2}}}+{\frac {d^{2}(q-q_{0})}{dz^{2}}}=4kf(q-q_{0})$

provided that, in $F$, we substitute for $q$ the value of $q-q_{0}$ resulting