Page:Scientific Memoirs, Vol. 1 (1837).djvu/478

(A)

${\begin{aligned}gv(\mathrm {q} &+{\bar {\omega }})\Sigma v_{\nu }({\bar {\omega }}+\mathrm {q} _{\nu })(1+\alpha r_{\nu })e^{-\alpha r_{\nu }}{\frac {\mathrm {x} _{\nu }-\mathrm {x} }{r_{\nu }^{3}}}\\&-(g-\gamma ){\bar {\omega }}v\Sigma {\bar {\omega }}_{\nu }v_{\nu }{\frac {\mathrm {x} _{\nu }-\mathrm {x} }{r_{\nu }^{3}}}=0\\gv(\mathrm {q} &+{\bar {\omega }})\Sigma v_{\nu }({\bar {\omega }}_{\nu }+\mathrm {q} _{\nu })(1+\alpha r_{\nu })e^{-\alpha r_{\nu }}{\frac {\mathrm {y} _{\nu }-\mathrm {y} }{r_{\nu }^{3}}}\\&-(g-\gamma ){\bar {\omega }}v\Sigma {\bar {\omega }}_{\nu }v_{\nu }{\frac {\mathrm {y} _{\nu }-\mathrm {y} }{r^{3}}}=0\\gv(\mathrm {q} &+{\bar {\omega }})\Sigma v_{\nu }({\bar {\omega }}_{\nu }+\mathrm {q} _{\nu })(1+\alpha r_{\nu })e^{-\alpha r_{\nu }}{\frac {\mathrm {z} _{\nu }-\mathrm {z} }{r_{\nu }^{3}}}\\&-(g-\gamma ){\bar {\omega }}v\Sigma {\bar {\omega }}_{\nu }v_{\nu }{\frac {\mathrm {z} _{\nu }-\mathrm {z} }{r_{\nu }^{3}}}=0,\end{aligned}}$

where the sums $\Sigma$ are to be extended to all the members $\nu$, that is to say, to all the molecules except that whose equilibrium we are considering.

7. The equations which we have just found are those which must take place in case of equilibrium, or in the natural state of a body composed of spherical molecules, if Franklin's hypothesis respecting statical electricity may be applied to the constitution of bodies also. The form in which the equations present themselves shows that this equilibrium takes place exactly as if there existed between each pair of molecules a reciprocal action, in the direction of the straight line which would join their centres of gravity, and would be represented by

(a)

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

Let us examine the nature of this action. We are able to distinguish in its expression the products $gv({\bar {\omega }}+\mathrm {q} ).v_{1}({\bar {\omega }}_{1}+\mathrm {q} _{1})$, $(g-\gamma ){\bar {\omega }}v.{\bar {\omega }}_{1}v_{1}$, the constant $\alpha$ and the variable $r_{1}$.

As the difference $(g-\gamma )$ between these two accelerative forces is to be supposed very small relatively to $g$, the product of this force by the masses $v({\bar {\omega }}+q)v_{1}({\bar {\omega }}_{1}+\mathrm {q} _{1})$ will, for a twofold reason, be greater than the product of the difference $g-\gamma$ by the masses ${\bar {\omega }}v.{\bar {\omega }}v_{1}$.

The value $\alpha$ depends on that of $f$ and $k$, that is to say on the repulsive force of the atoms of the æther, their mutual distances, their masses, and their volumes, which are all unknown to us. The agreement of the results of calculation with those of experiment requires that $\alpha$ should be a very high number.

On the condition that $a$ is very great, the first term of the expression (a) will decrease rapidly with $r_{1}$, because of the multiplier $e^{-\alpha r_{1}}$; if