Page:Scientific Memoirs, Vol. 2 (1841).djvu/477

changes to the extent of the entire tension situated at that place.

b. If any single place of the circuit is induced by any circumstance whatsoever to change its electric condition, all the other places of the circuit change theirs at the same time, and the same amount.

The constant $c$ is in the rule determined by ascertaining the electroscopic force at any place of the circuit. If, for instance, we designate by $u'$ the electroscopic force at a place of the circuit, the reduced abscissa of which is $y'$, then, in accordance with the general equation above stated,

$u'={\frac {A}{L}}y'-O'+c,$

where $O'$ represents the sum of the tensions abruptly passed over by the abscissa $y'$. If we now subtract this equation, valid for a certain place of the circuit, from the previous one belonging in the same manner to all places, we obtain

$u-u'={\frac {A}{L}}(y-y')-(O-O'),$

in which nothing more remains to be determined.

If the circuit, during its production, is exposed to no external deduction or adduction, the constant $c$ must be sought for in the circumstance that the sum of all the electricity in the circuit must be zero. This determination is founded on the fundamental position, that, from a previously indifferent state, both electricities constantly originate at the same time and in like quantity. To illustrate, by an example, the mode in which the constant $c$ is found in such a case, we will again consider the case treated of in § 16. In the portion $P$ of that circuit, $u$ is generally $={\frac {A}{L}}y+c$, where $y={\frac {x}{\chi \omega }}$, and in the portion $P'$ we have constantly $u={\frac {A}{L}}y-a'+c$, where $y={\frac {x-l}{\chi '\omega '}}=\lambda$. Since now the magnitude of the element, in the portion $P$, is $\omega \,dx$ or $\chi \omega ^{2}\,dy$, but in the portion $P'$ is $\omega '\,dx$ or $\chi '\omega ^{\prime 2}\,dy$, we obtain for the quantity of electricity contained in an element of the first portion

$\chi \omega ^{2}\,dy\left({\frac {A}{L}}y+c\right),$

and for the quantity contained in an element of the second portion