Page:Scientific Memoirs, Vol. 2 (1841).djvu/426

reduced abscisscæ and represent them generally by $y$, we obtain

${\begin{aligned}&XY={\frac {A}{L}}\cdot y\\-{}&X'Y'={\frac {A}{L}}\cdot y-a\\&X''Y''={\frac {A}{L}}\cdot y-(a+a'),\end{aligned}}$

and it is evident that $L$ is the same in reference to the whole length $AD$ or $FM$ as $y$ is to the lengths $FX$, $FX'$, $FX''$, on account of which $L$ is termed the entire reduced length of the circuit. Further, if we consider that for the abscissa corresponding to the ordinate $XY$ the tension has experienced no abrupt change, but that for the abscissa corresponding to the ordinate $X'Y'$ the tension has experienced the abrupt changes $a$, $a'$; and if we represent generally by $O$ the sum of all the abrupt changes of the tensions for the abscissa corresponding to the ordinate $y$, then all the values found for the various ordinates are contained in the following expression:

${\frac {A}{L}}\cdot y-O$.

But these ordinates express, when an arbitrary constant, corresponding to the length $AF$, is added to them, the electric forces existing at the various parts of the ring. If therefore we represent the electric force at any place generally by $u$ we obtain the following equation for its determination:

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

in which $c$ represents an arbitrary constant. This equation is generally true, and may be thus expressed in words: The force of the electricity at any place of a galvanic circuit composed of several parts, is ascertained by finding the fourth proportional to the reduced length of the entire circuit, the reduced length of the part belonging to the abscissa, and the sum of all the tensions, and by increasing or diminishing the difference between this quantity and the sum of all the abrupt changes of tension for the given abscissa by an arbitrary quantity which is constant for all parts of the circuit.

When the determination of the electric force at each place of the circuit has been effected, it only remains to determine the magnitude of the electric current. Now in a galvanic circuit of