Page:Scientific Memoirs, Vol. 2 (1841).djvu/443

circuit is given by the quotient of the difference between the electrical forces existing at the extremities of the portion and its reduced length. It is true, this rule was only advanced above for the case in which the circuit nowhere divides into several branches; but a very simple consideration, analogous to the one then made, derived from the equality of the abducted and adducted quantity of electricity in all sections of each prismatic part, is sufficient to prove that the same rule holds good for every single branch in case of a division of the circuit. Let us suppose that the circuit be divided, for instance, into three branches, whose reduced lengths are $\lambda$, $\lambda '$, $\lambda ''$; and, moreover, that at each of these places the undivided circuit and the single branches possess equal electrical force, and consequently no tension occurs there, and designate by $\alpha$ the difference between the electrical forces at these two places; then, according to the above rule, the magnitude of the current in each of the three branches is

${\frac {\alpha }{\lambda }},\quad {\frac {\alpha }{\lambda '}},\quad {\frac {\alpha }{\lambda ''}};$

whence it directly follows that the currents in the three branches are inversely as their reduced lengths; so that each separate one may be found when the sum of all three together is known. But the sum of all three is evidently equal to the magnitude of the current at any other place of the non-divided portion of the circuit, for otherwise the permanent state of the circuit, which is still constantly supposed, would not be maintained. If we connect with this the conclusion resulting from the above considerations, namely, that the magnitude of the current, and the nature of each homogeneous part of the circuit, give the dip of the corresponding straight line, representing the separation of the electricity, we are certain that the figure of the separation belonging to the non-divided portion of the circuit must remain the same so long as the current in it retains the same magnitude, and vice versâ; whence it follows that the variability of the current in the non-divided portion necessarily supposes that the difference between the electrical forces at the extremities of this portion is constant. If we now imagine, instead of the separate branches, a single conductor of the reduced length $A$ brought into the circuit which does not at all alter the magnitude of its current and its tensions, then, according to what has just been stated, the difference between the electrical forces