Page:Scientific Memoirs, Vol. 1 (1837).djvu/474

462 The expression for $$F$$ will then be reduced to All the quantities $$T_n$$ and $$V_n$$ being null, except $$T_0$$ and $$V_0$$, the values of $$Q_n$$ will also be null, except that of $$Q_0$$: the formula (2)′ will then give $$q - q_0 = \frac$$

When $$r = \infty$$ we must have $$q = q_0$$; we must then also have $$T_0 =0$$, and there will remain only $$q = q_0 + \frace^{-\alpha r}$$.

By performing the integrations of the formula (5)′ within the limits indicated, and observing that $$T_0 = 0$$, we shall obtain As, in the differential expression for $$F$$, we may change $$x'$$ into $$x' - \mathrm{x}$$, and x into $$x - \mathrm{x}$$, without any change taking place in its value, and as a similar change may be made in respect to the other coordinates, it follows that, by taking the point $$\mathrm{x}$$, $$\mathrm{y}$$, $$\mathrm{z}$$, as the origin of the coordinates, we shall be able, in the two preceding formulas, to put or, generally,

Now if, by placing the origin of the coordinates in the centre of each molecule respectively, we substitute these expressions of $$F$$ and $$q$$, and that previously found for $$G$$ in the equation (III)′, and take successively for $$V_0$$ as many constants as there are molecules, we shall find that the equation will be satisfied by taking for each molecule

By substituting for $$V_0^{(\nu)}$$ the value just found, we shall finally have