Page:Scientific Memoirs, Vol. 2 (1841).djvu/441

preceding consideration. For if we denote by $A$ the sum of the tensions, and by $L$ the reduced length of any galvanic circuit,

${\frac {A}{L}}$

expresses the magnitude of its current. If we now imagine a multiplier of $n$ similar convolutions each of the reduced length $\lambda$,

${\frac {A}{L+n\lambda }}$

indicates the magnitude of the current when the multiplier is brought into the circuit as an integral part. Moreover, if we grant, for the sake of simplicity, that each of the $n$ convolutions exerts the same action on the magnetic needle, the action of the multiplier on the magnetic needle is evidently

${\frac {nA}{L+n\lambda }}$,

when the action of an exactly similar coil of the circuit, without the multiplier on the needle, is taken as

${\frac {A}{L}}$.

Hence it follows directly that the action on the magnetic needle is augmented or weakened by the multiplier, according as $nL$ is greater or smaller than $L+n\lambda$, i. e., according as $n$ times the reduced length of the circuit without the multiplier is greater or smaller than the reduced length of the circuit with the multiplier. Further, a mere glance at the expression by which the action of the multiplier on the needle has been determined, will show that the greatest or smallest action occurs as soon as $L$ may be neglected with reference to $n\lambda$, and is expressed by

${\frac {A}{\lambda }}$.

If we compare this extreme action of the multiplier with that which a perfectly similarly constructed convolution of the circuit without the multiplier produces, we perceive that both are in the same ratio to one another as the reduced lengths $L$ and $\lambda$, which relation may serve to determine one of the values when the others are known. The expression found for the extreme action of the multiplier shows that it is proportional to the tension of the circuit, and independent of its reduced length; conse-