Page:Scientific Memoirs, Vol. 2 (1841).djvu/473

$u={\frac {(a+a')\chi '\omega 'x}{\chi '\omega 'l+\chi \omega l'}}+c,$

and that in the part $P'$ by the equation

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

If we substitute $\lambda$ and $\lambda '$ for ${\frac {l}{\chi \omega }}$ and ${\frac {l'}{\chi '\omega '}}$, the following more simple form may be given to these equations:—

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

(L).

From the form of these equations it will be immediately perceived, that when the conductibility, or the magnitude of the section, is the same in both parts, the expressions for $u$ and $u'$ undergo no other change than that the letter representing the conductibility or the section entirely disappears.

17. We will now proceed to the consideration of a galvanic circuit, composed of three distinct parts $P$, $P'$, and $P''$, which case comprises the hydro-circuit.

If we represent by $u$, $u'$, $u''$ respectively the electroscopic forces of the parts $P$, $P'$, and $P''$, then, according to § 15, the case there mentioned being here thrice repeated, we have, in accordance with the equation (c) there found, with respect to the part $P$,

$u=fx+c,$

with respect to the part $P'$,

$u'=f'x+c',$

and with respect to the part $P''$,

$u''=f''x+c'',$

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikisource.org/v1/":): {\displaystyle f} , $f'$, $f''$, $c$, $c'$, $c''$ may represent any constant magnitudes remaining to be determined from the nature of the problem, and each equation has only so long any meaning as the abscissæ refer to that part to which the equations appertain. If we suppose the origin of the abscissæ at that extremity of the part $P$, which is connected with the part $P''$, and choose the direction of the abscissæ so that they lead from the part $P$ to that of $P'$, and from thence into $P''$; if we further respectively