Page:Scientific Memoirs, Vol. 1 (1837).djvu/466

454 which lead directly to the complete integral $$C$$ being an arbitrary constant.

In order to determine, by means of this equation, the density $$q$$, we must substitute for $$F$$, $$G$$, $$G_1$$, $$G_2$$, … $$G_\nu$$, &c. the integrals which they represent. If the rectangular co-ordinates are changed into polar co-ordinates by means of the known formulæ the expression for $$F$$ takes the form (see the additions to the Connaissance des Temps for the year 1829, p. 356)

The coefficient $$P_n$$ being given by the formula in which and the limits of the integrals relative to $$\theta'$$ and $$\psi'$$ should be such that the value of $$F$$ may take in the whole space, except the small portions occupied by the material molecules.

In order to have the expression for $$G$$, let us in like manner put and represent by $$\Pi_n$$ the function $$P_n$$, when $$r'$$, $$\theta'$$, $$\psi'$$, are therein changed into $$\rho$$, $$\omega$$, $$\phi$$. Then, if we suppose the origin of the co-ordinates to be taken in the interior of the molecule, we shall have (see Connaissance des Temps for the year 1829, p. 357)