Page:Scientific Memoirs, Vol. 2 (1841).djvu/513

part of the circuit, in which no chemical change occurs, are unalterable and given, it is, in accordance with the general equation given in § 18, which likewise holds for our present case, only requisite for the perfect knowledge of the function $u$, that we are able to determine the tensions and reduced lengths for each place within the extent in which the chemical change takes place.

But evidently the reduced length of the disc $M$ is

${\frac {dx}{\chi \omega }};$

or if we substitute for $\chi$ its value just found,

${\frac {a+(b-a)z}{ab\omega }}dx;$

we accordingly obtain the reduced length of any part of that extent, if we integrate the above expression, and take the limits of the integral corresponding to the commencement and end of the part. If now we bear in mind that the integral

$\int {\frac {a+(b-a)z}{ab\omega }}dx$

may also be written thus:

${\frac {l}{b\omega }}+{\frac {b-a}{ab\omega ^{2}}}\int z\omega dx,$

when $l$ represents the length of the part, over which the integral is to be extended, and $z\omega dx$ expresses merely the space which the constituent $A$ in the disc $M$ occupies; consequently $\int z\omega dx$, the sum of all the spaces which the constituent $A$ fills in the part whose reduced length has to be found, it is obvious that the reduced length of the entire portion, in the act of decomposition, remains unchangeable during the chemical change, since, as we have supposed, each constituent maintains, under all circumstances, constantly the same volume. The same result may also be directly deduced from what was advanced in the preceding paragraph; however, this unchangeability only relates to the reduced length of the entire portion; the reduced length of a part of it does not in general depend merely on the actual length of this part, but likewise on the contemporaneous chemical distribution of the constituents in the extent, and must therefore, in each separate case, be first ascertained in the manner indicated.