Page:Scientific Memoirs, Vol. 2 (1841).djvu/475

duced from the tensions, we now find, when $\lambda$, $\lambda '$, $\lambda ''$ are respectively substituted for ${\frac {l}{\chi \omega }}$, ${\frac {l'}{\chi '\omega '}}$, ${\frac {l''}{\chi ''\omega ''}}$,

${\begin{aligned}f&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot {\frac {1}{\chi \omega }},\\f'&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot {\frac {1}{\chi '\omega '}},\\f''&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot {\frac {1}{\chi ''\omega ''}}.\end{aligned}}$

and by the aid of these values we find further,

${\begin{aligned}c'&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\left({\frac {l}{\chi \omega }}-{\frac {l}{\chi '\omega '}}\right)-a'+c,\\c''&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot \left({\frac {l}{\chi '\omega '}}-{\frac {l+l'}{\chi ''\omega ''}}+{\frac {l}{\chi \omega }}\right)-(a'+a'')+c.\end{aligned}}$

By substituting these values, we obtain for the determination of the electroscopic force of the circuit in the parts $P$, $P'$, $P''$ respectively, the following equations:

$\left.{\begin{aligned}u&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot {\frac {x}{\chi \omega }}+c\\u'&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot \left({\frac {x-l}{\chi '\omega '}}+{\frac {l}{\chi \omega }}\right)-a'+c\\u''&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot \left({\frac {x-(l+l')}{\chi ''\omega ''}}+{\frac {l'}{\chi '\omega '}}+{\frac {l}{\chi \omega }}\right)-(a'+a'')+c\end{aligned}}\right\}$

(L′).

and it is easy to see, that these equations, with the omission of the letter $\chi$ or $\omega$ (both where they are explicit, as well as in the expressions for $\lambda$, $\lambda '$, $\lambda ''$), are the true ones for the case $\chi =\chi '$ , or $\omega =\omega '=\omega ''$.

18. These few cases suffice to demonstrate the law of progression of the formulæ ascertained for the electroscopic force, and to comprise them all in a single general expression. To do this with the requisite brevity, for the sake of a more easy and general survey, we will call the quotients, formed by dividing the length of any homogeneous part of the circuit by its power of conduction and its section, the reduced length of this part; and when the entire circuit comes under consideration, or a portion of it, composed of several homogeneous parts, we understand by its reduced length the sum of the reduced lengths of all its parts. Having premised this, all the previously found expressions for the electroscopic force, which are given by the equa-