Page:Scientific Memoirs, Vol. 2 (1841).djvu/512

nitude of its current, which results from the above-found formulæ, is

${\frac {A}{L+{\frac {v}{a\omega }}+{\frac {w}{b\omega }}}}.$

If now a conductor of the length $v+w$, and of the power of conduction $\chi$ with the same section, being taken instead of the two former, leaves the current of the circuit unchanged, then must

${\frac {v}{a\omega }}+{\frac {w}{b\omega }}={\frac {v+w}{\chi \omega }},$

whence we find

$\chi ={\frac {ab(v+w)}{bv+aw}}.$

But it is perfectly indifferent for the magnitude of the current, whether the entire length $v$ be situated near the entire length $w$, or any number of discs be formed of the two, which are arranged in any chosen order, if only the extreme parts remain of the same kind, as otherwise a change might result in the sum of the tensions, consequently also in the magnitude of the current. If we extend this law, which holds for every mechanical mixture, likewise to a chemical compound, the above value found for $\chi$ evidently gives the conducting power of the compound, where, however, it has been taken for granted that the two parts of the circuit, even after the mixture, still occupy the same volume, for $v$ and $w$ are here evidently proportional to the spaces occupied by the two bodies mixed with each other.

If we now apply this result to our subject, and therefore put, instead of $v$ and $w$, the values $z$ and $1-z$, which express the relations of space of the two constituents in the disc $M$, we obtain, when $a$ denotes the conducting power of the one constituent $A$, and $b$ the same for the constituent $B$; further, $\chi$ the power of conduction of the mixture of the two contained in the disc $M$, the following expression for $\chi$,

$\chi ={\frac {ab}{a+(b-a)z}}.$

37. Having thus determined the power of conduction at each place of the extent in the act of decomposition, there only remains to be ascertained the nature of the function $u$ at each such place; and since all tensions and reduced lengths in the