Page:Scientific Memoirs, Vol. 2 (1841).djvu/517

and determine in accordance with this statement the constant $c$, our last equation acquires the following form:—

${\frac {\Sigma +\Omega z}{a+(b-a)z}}={\frac {\Sigma +\Omega \zeta }{a+(b-a)\zeta }}\cdot e^{{\frac {(b-a)\Sigma -a\Omega }{ab\omega (n-m)}}(\chi -x,)}$

where $e$ denotes the base of the natural logarithms. The following consideration leads to the determination of the value $\chi$. Since, namely, $\zeta$ represents the space which the constituent $A$ occupies in each individual disc of the changeable portion previous to the commencement of the chemical decomposition, if we denote by $l$ the actual length of this portion, $l\zeta$ expresses the sum of all the spaces which the constituent $A$ occupies on the entire expanse of the changeable portion; but this sum must constantly remain the same, since, according to our supposition, no part of either of the constituents is removed from this portion, and both maintain, under all circumstances, the same volume, even after chemical decomposition has taken place; we obtain, therefore,

$l\zeta =\int z\;dx,$

where for $z$ is to be substituted its value resulting from the previous equation, and the abscissæ corresponding to the commencement and end of the changeable portion are to be taken as limits of the integral.

These two last equations, in combination with that found at the end of the previous paragraph, answer all questions that can be brought forward respecting the permanent state of the chemical alteration, and the change in the electric current thus produced, and so form the complete base to a theory of these phænomena, the completing of the structure merely awaiting a new supply of materials from experiment.

40. At the conclusion of these investigations we will bring prominently forward a particular case, which leads to expressions that, on account of their simplicity, allow us to see more conveniently the nature of the changes of the current produced by the chemical alteration of the circuit. If, for instance, we admit $a=b$, and $\alpha =\beta$, the differential equation obtained in the preceding paragraph changes into the following:

$0=\Sigma \;dx-a\omega (n-m)\;dz,$

whence we obtain by integration