Page:Scientific Memoirs, Vol. 2 (1841).djvu/455

constant at all times, it merely depends on the magnitude of the instant of time $d\,t$, we can consequently extend it to the unity of time; if we place the present constant difference of the forces $u'-u$ equal to the unity of force, it then becomes

$\alpha mm'$.

This quantity of electricity is for the two elements $E$ and $E'$ whose position is invariable, constant under the same circumstances, on which account it may be employed in the determination of the power of conduction just mentioned. For if we understand by $\varsigma$ the quantity of electricity transferred from $E'$ to $E$ in the unity of time, with a constant difference of the electroscopic forces equal to the unity of force, we have

$\varsigma =\alpha mm'$,

and then

$\chi =\alpha mm's$.

If we take from this last equation the value of $\alpha mm'$ and substitute it in the expression

$\alpha mm'(u'-u)\,d\,t$,

we obtain for the variable quantity of electricity which passes over in the instant of time $d\,t$ from $E'$ to $E$, the following:

${\frac {\chi (u'-u)\,d\,t}{s}}$,

(♂)

which expression is not accompanied by the above-mentioned disproportion between the members of the differential equation, as will soon be perceived.

7. The course hitherto pursued was based upon the supposition that the action exerted by one element on the other is proportional to the product of the space occupied by the two elements, an assumption which, as was already observed in § 4, can no longer be allowed in cases where it is a question of the mutual action of elements situated indefinitely near each other, because it either establishes a relation between the magnitudes of the elements and their mutual distances, or prescribes to these elements a certain form. The previously found expression (♂) for the variable quantity of electricity passing from one element to the other, possesses therefore no slight advantage in being entirely independent of this supposition; for whatever may have to be placed in any determinate case instead of the product $mm'$, the expression (♂) constantly remains the