Page:Scientific Memoirs, Vol. 1 (1837).djvu/638

In the same manner we find, if three series have not a more powerful action than two

$\delta ={\frac {1}{2}}(\gamma =\beta ),$

i. e. this happens when the length of the free ends is half as great as half the sum of the differences between the length of the first series and the lengths of the second and third series.

Having thus proved that by increasing the series of convolutions we never obtain a maximum of the electric current, and therefore that a greater increase would only do harm, we proceed to the general consideration of the subject.

We therefore suppose the convolutions of a series of the bespun metallic wire to lie thick on one another. Let the length of the space on which the convolutions may be wound up be ${}=\alpha$, the thickness of the wire ${}=\beta$; let the thickness of the wire covered with silk surpass the thickness of the uncovered wire by the excess $\beta$, so that it be ${}=b+\beta$,

the length of a convolution be ${}=c$, the lengths of the free ends of the wire ${}=m$; the number of convolutions then which can be wound in one series upon the armature is ${}={\frac {\alpha }{b+\beta }}$ and the length of the wire of these convolutions ${}={\frac {\alpha }{b+\beta }}\cdot c$, and the whole length which the electricity has to run through for one series of convolutions

${}={\frac {\alpha }{(b+\beta )}}c+m.$

If we assume the resistance offered by a wire of the same substance, whose length ${}=1$, and whose thickness ${}=1$, as unity, the resistance for one series of convolutions becomes

${\frac {={\frac {\alpha }{b+\beta }}c+m}{b^{2}}}={\frac {ac+(b+\beta )m}{b^{2}(b+\beta )}}.$

Further, let the electromotive power produced in one convolution, which, according to the second and third of our laws above proved, remains the same for every magnitude of the convolutions and for every thickness of the wire, be called $f$; the electromotive power produced in a series of convolutions is therefore, according to the first of the above laws,

${}={\frac {\alpha }{b+\beta }}\cdot f,$

and consequently the power of the electric current for a series of convolutions, or

$p_{1}={\frac {\alpha b^{2}f}{\alpha c+(b+\beta )m}}.$

We must now for our purpose express the length of a convolution or $c$