Page:Scientific Memoirs, Vol. 1 (1837).djvu/471

$=\Sigma _{0}^{\infty }\iint f{\bigg \lbrace }{\frac {1}{r^{''+1}}}\int _{0}^{r}\Sigma _{0}^{\infty }\Omega _{i}^{'}(r')V_{i}^{'}r^{'n+1}\,dr'+r^{n}\int _{r}^{\infty }\Sigma _{0}^{\infty }{\frac {\Omega _{i}^{'}(r')V_{i}^{'}\,dr'}{r^{'n}}}{\bigg \rbrace }P_{n}\sin \theta '\,d\theta '\,d\psi '.$

The functions $T'$, $V_{i}^{'}$ of this expression remain arbitrary; and, as the sum of an infinite number of these fund ions may be employed to represent any function whatsoever, they will serve as two arbitrary functions which are to complete the integral of the equation (1).

When in some particular cases the integrations of the preceding formula shall have been performed by substituting its expression in the equation (III), the functions $T_{i}$ and $V_{i}$ will be determined by comparing them with those of the same order introduced by means of the different expressIons for $G$; so that this equation may become identical. All being thus determined, the densty $q$ given by the formula (2) will be known.

We have hitherto left our formula in all their generality, so that one may be the better able to judge of the restrictions to which we shall subject them while making the first applications of them. In the present state of our physical knowledge, the figure of the material molecules is totally unknown. We will therefore begin by considering the most simple case,—that in which their form is spherical, and their density uniform. We will, besides, assign to these molecules a very small volume, and suppose them in their state of equilibrium at a mutual distance, which is very considerable as compared with their dimensions. This manner of considering the constitution of bodies has been adopted by several philosophers as that which is most conformable to truth, and presents at the same time a considerable advantage in an analytical point of view. In adopting it we shall be able, by approximation, to consider the æther as if it were continuously diffused in all directions; and to disregard, in the integration of the formula (5), the small spaces occupied by the material molecules. But as, by proceeding in this manner, we should include in the repulsion of the æther a surplus which is due to the actions answering to the points of space which are really occupied by the molecules, we shall compensate for this surplus by adding to the action of each molecule an action equal and contrary to that of a quantity of æther of the same volume as the molecule, and of the same density as that which answers to the point of space which the molecule occupies. This is done by substituting $g{\bar {\omega }}+f\mathrm {q}$ for $g{\bar {\omega }}$ in the expression for $G$ ($\mathrm {q}$ representing the density which the æther would have at the point occupied by the molecule, and within so small a space we will suppose that density constant), and by extending the integrals