Page:Scientific Memoirs, Vol. 1 (1837).djvu/479

then $r_{1}$ has a greater value than that which renders this expression null, the force represented by the last term will preponderate over that represented by the first; and if $r_{1}$ be so great that this term may be neglected as of no value, then the only remaining force will be that given by the last term. This term being negative, the force which corresponds with it tends to bring the molecules nearer to each other; and as it is in the direct ratio of the product of the masses, and the inverse ratio of the square of the distance, it will exactly represent the universal gravitation which takes place at finite distances.

By diminishing $r_{1}$ we shall obtain a value that will satisfy the equation

(b)

$gv({\bar {\omega }}+\mathrm {q} )v_{1}({\bar {\omega }}_{1}+\mathrm {q_{1}} ){\frac {(1+\alpha r_{1})e^{-\alpha r_{1}}}{r_{1}^{2}}}-(g-\gamma )v{\bar {\omega }}.v_{1}{\bar {\omega }}_{1}{\frac {1}{r_{1}^{2}}}=0.$

At this distance two molecules would remain in equilibrium, and as the differentiation of this equation gives the result

$-gv({\bar {\omega }}+\mathrm {q} )v_{1}({\bar {\omega }}+\mathrm {q} _{1}){\frac {\alpha ^{2}e^{-\alpha r_{1}}}{r_{1}}}$

which is always negative, the equilibrium will be permanently fixed. Should it be attempted, by the application of an external force, to bring the molecules nearer to each other, the repulsive force represented by the first term of the expression (a), which would now increase in a greater ratio than the attractive force represented by the last term, would produce a resistance to such an approximation: on the other hand, if it should be sought to remove the molecules to a greater distance from each other, the repulsive force would decrease in a greater ratio, and the attractive would preponderate and prevent the separation. These two molecules would therefore be so placed relatively to each other as by mutual adhesion to form a whole, and we should not be able to remove the one without at the same time removing the other. Thus these molecules present a picture in which the hooked atoms of Epicurus are as it were generated by the love and hatred of the two different matters of Empedocles.

As the attractive force is null at the distance which we have been just now considering, and at a greater distance decreases as the square of the distance of the molecules, there must be an intermediate point at which it reaches its maximum. By the ordinary rules of the differential calculus we find that the function (a) is a maximum when

(c)

${\begin{aligned}-&gv({\bar {\omega }}+\mathrm {q} )v_{1}({\bar {\omega }}_{1}+\mathrm {q} _{1})(1+\alpha r_{1}+{\frac {1}{2}}\alpha ^{2}r_{1}^{2})e^{-\alpha r_{1}}+(g-\gamma )\\&v{\bar {\omega }}.v_{1}{\bar {\omega }}=0;\end{aligned}}$

that is to say, that it is at the distance $r_{1}$ we should find, by the resolu-