Page:Scientific Memoirs, Vol. 2 (1841).djvu/472

places of contact, by the springing over of which, in the direction of the abscissæ, we arrive from the greater to the smaller electroscopic force, are to be regarded as positive, in the contrary case as negative, where, however, it must not be overlooked that every positive force has to be taken as greater than every negative, and the negative as greater than the actually smaller), we obtain

$a=f'(l+l')+c'-c,$

and

$a'=fl-f'l+c-c',$

whence directly results

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

But now at each of the places of contact when $\chi$ and $\omega$ represent the power of conduction and the section of the part $P$, and $\chi '$ and $\omega '$ the same for $P'$, in accordance with the considerations developed in § 13, there arises the conditional equation

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

where $\left({\frac {du}{dx}}\right)$ and $\left({\frac {du'}{dx}}\right)$ represent the values of ${\frac {du}{dx}}$ and ${\frac {du'}{dx}}$ at the place of contact. From the equations at the commencement of this paragraph for the determination of the electroscopic force in each single part of the circuit, we, however, obtain the value of $x$ to be allowed to each,

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

which converts the conditional equation in question into

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

From this, and the equation $a+a'=fl+f'l'$ just deduced from the tensions, we now find the values of $f$ and $f'$ thus:

${\begin{aligned}f&={\frac {(a+a')\chi '\omega '}{\chi '\omega 'l+\chi \omega l'}},\\f'&={\frac {(a+a')\chi \omega }{\chi '\omega 'l+\chi \omega l'}},\end{aligned}}$

and with the help of these values we find

$c'=c-a'+{\frac {(a+a')(\chi '\omega 'l-\chi \omega l')}{\chi '\omega 'l+\chi \omega l'}}.$

Hence the electroscopic force of the circuit in the part $P$ is expressed by the equation