needle; if we imagine them, therefore, so arranged near one another, that though they are separated by a non-conducting layer, they are yet situated so close together that the position of each one toward the magnetic needle may be regarded as the same, they would produce a greater effect on the magnetic needle in proportion as their number increased. Such an arrangement is termed a multiplier.
Now, let be the sum of the tensions of any circuit, and its reduced length; let also be the reduced length of one of the interposed conductors formed into a multiplier of convolutions; then, if we represent the reduced length of one such convolution by , , the action of the multiplier on the magnet needle will be proportional to the value
But the action of a similar coil of the circuit, without the multiplier, is, according to the same standard,
and we will suppose the portion of the circuit, whence the coil is taken, to be of the same nature as in the multiplier; accordingly the difference between the former and the present effect is
which is positive or negative according as is greater or less than . Consequently the action on the magnetic needle will be augmented or diminished by the multiplier formed of coils, according as the times reduced length of the circuit, without interposed conductor, is greater or less than the entire reduced length of the circuit with the interposed conductor.
If is incomparably greater than , the action of the multiplier on the needle will be
To this value, which indicates the extreme limit of the action by means of the multiplier, whether it be strengthening or weakening, belong several remarkable properties, which we will briefly notice. It is constantly supposed that the multiplier is formed of so many coils that the magnitude of its action may,
