Page:Scientific Memoirs, Vol. 1 (1837).djvu/480

tion of this equation, that the molecules attract each other most forcibly.

Recapitulating these results, we shall say then, that the action of two spherical molecules on each other is repulsive, from their point of contact to a distance given by the equation (b). At this distance the two molecules are in a state of fixed equilibrium, and as it were linked together; at a greater distance their action is attractive, and the attraction continues to increase until they are at the distance $r_{1}$ furnished by the equation (c), which distance is still very inconsiderable because of the magnitude of $\alpha$ in the exponential term $e^{-\alpha r_{1}}$. From this point the force remains always attractive, and, when the distance has acquired a sensible value, follows the inverse ratio of the square of the distance. All these properties of molecular action flow as necessary consequences from Franklin's hypothesis respecting statical electricity, and appear perfectly conformable to those indicated by the phænomena.

Let us suppose four homogeneous and equal molecules placed at the points of a regular tetrahedron. If we assume as the origin of the coordinates the place occupied by the molecule whose equilibrium it is proposed to consider, and as the plane of the $x\,y$, a plane parallel to that in which the three others are found, the coordinates of these molecules will be given by the formulæ

${\begin{aligned}\mathrm {x} &=0&\mathrm {y} &=0&\mathrm {z} &=0\\\mathrm {x} _{1}&={\frac {r}{\sqrt {3}}}\cos \beta &\mathrm {y} _{1}&={\frac {r}{\sqrt {3}}}\sin \beta &\mathrm {z} _{1}&=r{\sqrt {\frac {2}{3}}}\\\mathrm {x} _{2}&={\frac {r}{\sqrt {3}}}\cos \left(\beta +{\frac {2\pi }{3}}\right)&\mathrm {y} _{2}&={\frac {r}{\sqrt {3}}}\sin \left(\beta +{\frac {2\pi }{3}}\right)&\mathrm {z} _{2}&=r{\sqrt {\frac {2}{3}}}\\\mathrm {x} _{3}&={\frac {r}{\sqrt {3}}}\cos \left(\beta +{\frac {4\pi }{3}}\right)&\mathrm {y} _{3}&={\frac {r}{\sqrt {3}}}\sin \left(\beta +{\frac {4\pi }{3}}\right)&\mathrm {z} _{3}&=r{\sqrt {\frac {2}{3}}}\end{aligned}}$

where $r$ denotes the mutual distance of the molecules, which is the same for all; $\beta$ the angle which is formed in the plane of $x\,y$ with the axis of the $x$, by the projection of the straight line drawn from the molecule placed at the origin of the coordinates to the first of the three others; and $\pi$ the semicircumference.

If these values be substituted in the three equations (A), and it is observed that we always have, whatever may be the value of $\beta$,

${\begin{aligned}\cos \beta &+\cos \left(\beta +{\frac {2\pi }{3}}\right)+\cos \left(\beta +{\frac {4\pi }{3}}\right)=0,&\sin \beta &+\sin \left(\beta +{\frac {2\pi }{3}}\right)\\&+\sin \left(\beta +{\frac {4\pi }{3}}\right)=0,\end{aligned}}$

it will be seen that the two first are verified by themselves, and that