Page:Scientific Memoirs, Vol. 2 (1841).djvu/510

tive; the sign of the function $\phi$, therefore, changes when the direction of the decomposition is transposed from the one constituent to the other. The nature of the function $\phi$ is as little known to us as the size and form of the elements on which it is dependent; nevertheless, we may, in our inquiries, regard its absolute value as constant, since the size and form of the corporeal particles, acting on each other, must be conceived to be unchangeable so long as the two constituents remain the same, and the supposition that the two constituents constantly maintain for every chemical equivalent the same volume, renders attention to the mutual distance of the chemically different particles unnecessary, as regard has already been paid, when determining the electroscopic forces in the disc $\mathrm {M}$, to the relative distances of the elements of each constituent.

35. To determine the magnitude of the reaction $\mathrm {Y}$, which in the disc $\mathrm {M}$ opposes the latent electricity of the neighbouring discs to the decomposing force, we have nothing further to do than to substitute in the expression for $\mathrm {Z}$ instead of $u$, the sum of all the latent electroscopic forces in the disc $\mathrm {M}$. Since now the sum of these latent forces is $mz+n(1-z)$, we obtain for the determination of the force $\mathrm {Y}$, which is called into existence by the change in the chemical equivalent of the constituents, and which opposes the decomposition, after due determination of its sign, the following equation:

$\mathrm {Y} =4\chi \omega {\frac {dz}{dx}}.i(n-m).z(1-z).{\frac {m\beta -n\alpha }{\alpha z+\beta (1-z)}}$.

If now we substitute for $\mathrm {X}$, $\mathrm {Y}$, and $\mathrm {Z}$ the values found in the equation

$\mathrm {X} +\mathrm {Y} =\mathrm {Z}$,

we obtain, after omitting the common factor $4z(1-z)$, and multiplying the equation by ${\frac {\alpha z+\beta (1-z)}{i(m\beta -n\alpha }}$, as the condition of the permanent state is the chemical equivalent of the two constituents, the equation

${\begin{aligned}0&=\chi \omega {\frac {du}{dx}}+{\frac {\phi mn}{i(m\beta -n\alpha )}}\times .[\alpha z+\beta (1-z)]\omega \\&-\chi \omega (n-m){\frac {dz}{dx}},\end{aligned}}$

which, when we put