Page:Scientific Memoirs, Vol. 2 (1841).djvu/518

$z-\zeta ={\frac {\Sigma (x-\chi )}{a\omega (n-m)}},$

when $\chi$ designates the value of $x$, for which $z=\zeta$. Since in this case the value of $z$ constantly changes to the same amount on like differences of the abscissæ, the abscissa $\chi$, which belongs to its mean value $\zeta$, as it was at all places of the changeable portion previous to the commencement of the chemical decomposition, must be referred to the middle of this portion. If, therefore, $z'$ and $z''$, as above, represent the values of $z$, which correspond to the commencement and end of the variable portion, and $l$ the actual length of this portion, it follows, from our last equation, that

$z''-\zeta =+{\frac {1}{2}}{\frac {l\Sigma }{a\omega (n-m)}},$

and

$z'-\zeta =-{\frac {1}{2}}{\frac {l\Sigma }{a\omega (n-m),}}$

and from these two equations results

$(n-m)(z''-z')={\frac {l}{a\omega }}\cdot \Sigma ;$

or, if we put, instead of ${\frac {l}{a\omega }}$, by which here nothing further is expressed than the unchangeable reduced length of the chemically variable portion, the letter $\lambda$, the following:

$(n-m)(z''-z')=\lambda \Sigma .$

If we place this value of $(n-m)(z''-z')$ in the equation found in § 38,

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

and at the same time substitute for $\Sigma$ its value $S'+\psi \omega \alpha$, we obtain

$S'=S-{\frac {\Phi \lambda }{L}}(S'+\psi \omega \alpha ),$

an equation, the form of which is extremely well suited to indicate in general the nature of the change of the current produced by the chemical alteration, and the expressions of which coincide exceedingly well with the numerous experiments I have made on the fluctuation of the force in the hydro-circuit, and of which only a small part have been published^[1].

1. Schweigger's Jahrbuch, 1825, Part 1 and 1826, Part 2.