Page:Scientific Memoirs, Vol. 2 (1841).djvu/478

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

If we now integrate the first of the two preceding expressions from $y=O$ to $y=\lambda$, we then obtain for the whole quantity of electricity contained in the part $P$,

$\chi \omega ^{2}\left[{\frac {A}{2L}}\lambda ^{2}+c\lambda \right];$

in the same manner we obtain, by integrating the second expression from $y=\lambda$ to $y=\lambda +\lambda '$, for the entire quantity of electricity contained in the portion $P'$

$\chi '\omega ^{\prime 2}\left[{\frac {A}{2L}}(\lambda ^{\prime 2}+2\lambda \lambda ')-a'\lambda '+c\lambda '\right].$

But the sum of the two last found quantities must, in accordance with the above-advanced fundamental position, be zero. We thus obtain the equation required for the determination of the constant $c$, and it only remains to be observed that $\lambda$ and $\lambda '$ are the reduced lengths corresponding to the portions $P$ and $P'$.

We have hitherto always tacitly supposed only positive abscissæ. But it is easy to be convinced that negative abscissæ may be introduced quite as well. For let $-y$ represent such a negative reduced abscissa for any place of the circuit, then $L-y$ is the positive reduced abscissa pertaining to the same place, for which the general equation found is valid; we accordingly obtain

$u={\frac {A}{L}}(L-y)-O+c$

or

$u=-{\frac {A}{L}}y-(O-A)+c.$

But $O-A$ evidently expresses, if regard be had to the general rule expressed in § 16, the sum of all the tensions abruptly passed over by the negative abscissa, whence it is evident that the equation still retains entire its former signification for negative abscissæ.

19. If we imagine one of the parts of which the galvanic circuit is composed to be a non-conductor of electricity, i. e. a body whose capacity of conduction is zero, the reduced length of the entire circuit acquires an indefinitely great value. If we now make it a rule never to let the abscissæ enter into the non-conducting part, in order that the reduced abscissa $y$ may constantly retain a finite value, the general equation changes into the following: