Page:Scientific Memoirs, Vol. 2 (1841).djvu/516

reduced length of the entire circuit, which, as we have seen, remains under all circumstances the same, it results

$S'={\frac {\mathrm {A} -\Phi (n-m)(z''-z')}{\mathrm {L} }}$;

or, if we write for ${\frac {\mathrm {A} }{\mathrm {L} }}$ its equivalent $\mathrm {S}$,

$S'=S-{\frac {\Phi (n-m)(z''-z')}{\mathrm {L} }},$

so that, therefore, ${\frac {\Phi (n-m)(z''-z)}{\mathrm {L} }}$ designates the decrease produced in the magnitude of the current by the chemical alteration.

39. After all these intermediate considerations, we now proceed to the final determination of the chemical alteration in the changeable portion, and the change of the current in the whole circuit produced by this chemical alteration, where, however, we have constantly to keep in view only the permanent state of the altered portion. If we substitute in the equation (♁) given in §35, for $\chi \omega {\frac {du}{dx}}$ its value $\mathrm {S} '$, which, as we have just found, is solely dependent on the fixed and unalterable values of $z$, and therefore has to be treated in the calculation as a constant magnitude; further, for $\chi$ its value ${\frac {ab}{a+(b-a)z}}$, given in §36, this equation changes into

$0=S'+\psi \omega \beta +\psi \omega (\alpha -\beta )z-{\frac {ab\omega (n-m)}{a+(b-a)z}}\cdot {\frac {dz}{dx}}$;

or if we place $S'+\psi \omega \beta =\Sigma$, and $\psi \omega (\alpha -\beta )=\Omega$, into

$0=\Sigma +\Omega z-{\frac {ab\omega (n-m)}{a+(b-a)z}}\cdot {\frac {dz}{dx}}$,

from which, by integration, we deduce the following:

$c={\frac {(b-a)\Sigma -a\Omega }{ab\omega (n-m)}}x+\log {\frac {\Sigma +\Omega z}{a+(b-a)z}}$,

where $c$ represents a constant remaining to be determined. If we designate by $\chi$ the abscissa of that place of the chemically changed portion for which $z$ has still the same value, which, previous to the commencement of the chemical decomposition, belonged to each place of this portion, for which therefore $z=\zeta$,