Page:Scientific Memoirs, Vol. 2 (1841).djvu/474

designate by $l$, $l'$, and $l''$ the lengths of the parts $P$, $P'$, $P''$; and lastly, let $u''_{2}$ and $u_{1}$ represent the values of $u''$ and $u$ at the place of contact where $x=0$, and $u_{2}$ and $u'_{1}$ the values of $u$ and $u'$ at the place of contact where $x=l$, and $u'_{2}$ and $u''_{1}$ the values of $u'$ and $u''$ at the place of contact where $x=l+l'$, then we obtain

${\begin{aligned}u''_{2}&=f''(l+l'+l'')+c''&u_{1}&=c\\u_{2}&=fl+c&u'_{1}&=f'l+c'\\u'_{2}&=f'(l+l')+c'&u''_{1}&=f''(l+l')+c''.\end{aligned}}$

If we call $a$ the tension which occurs at the place of contact where $x=0$, $a'$ that at the place of contact where $x=l$, and $a''$ that at the place of contact where $x=l+l'$, we obtain, if we pay due attention to the general rule stated in the preceding paragraph,

${\begin{aligned}a&=f''(l+l'+l'')+c''-c\\a'&=fl-f'l+c-c'\\a''&=f'(l+l')-f''(l+l')+c'-c'',\end{aligned}}$

and hence

$a+a'+a''=fl+f'l'+f''l''.$

But from the considerations developed in § 13, when $\chi$ and $\omega$ represent the power of conduction and the section for the part $P$, $\chi '$ and $\omega '$ the same for the part $P'$, and $\chi ''$ and $\omega ''$ for the part $P''$, at the individual places of contact, the following conditional equations are obtained:

$\chi \omega \left({\frac {du}{dx}}\right)=\chi '\omega '\left({\frac {du'}{dx}}\right)=\chi ''\omega ''\left({\frac {du''}{dx}}\right),$

where $\left({\frac {du}{dx}}\right)$, $\left({\frac {du'}{dx}}\right)$, $\left({\frac {du''}{dx}}\right)$ represent the particular values of ${\frac {du}{dx}}$, ${\frac {du'}{dx}}$, ${\frac {du''}{dx}}$, belonging to the places of contact. From the equations stated at the commencement of the present paragraph for the determination of the electroscopic force in the single parts of the circuit, we obtain for every admissible value of $x$,

${\frac {du}{dx}}=f,\qquad {\frac {du'}{dx}}=f',\qquad {\frac {du''}{dx}}=f'',$

by which the preceding conditional equations are converted into

$\chi \omega f=\chi '\omega 'f'=\chi ''\omega ''f''.$

From these, and the equation between $f$, $f'$, and $f''$ above de-