Page:Scientific Memoirs, Vol. 1 (1837).djvu/642

maximum of the current at somewhat about thirteen series of convolutions, and the current then becomes about 115 times stronger than when produced by one convolution.

We will here separately consider the case in which $m=0$, i. e. where the spirals have no free ends, but close in themselves. If we put $m=0$ in the expression of the current for one convolution, for one series of convolutions, and for $n$ series of convolutions, we shall then obtain

for a single convolution ${}={\frac {b^{2}f}{2q\pi +\pi (b+\beta )}}$

for a series of convolutions ${}={\frac {b^{2}f}{2q\pi +\pi (b+\beta )}}$

for $n$ series of convolutions ${}={\frac {b^{2}f}{2q\pi +n\pi (b+\beta )}}$

whence it follows that here the current in one convolution is just as strong as in a series consisting of any number of convolutions; and that in both these cases it is stronger than when several series of convolutions cross one another (for $n$ is quite a positive number). The expression of the current for a convolution may moreover be exhibited thus

${\frac {f}{\frac {(2q+b+\beta )\pi }{b^{2}}}}$

i. e. it is equal to the electromotive power, divided by the resistance offered by a convolution; and in effect it is evident that in this case of $m=0$ a series of convolutions must act just in the same manner as a single convolution; for with the increase of the number of convolutions the electromotive power and the resistance become increased in the same proportion, consequently the quotient of the one by the other, or the electric current remains unchanged. It is also now evident that in effect a second series of convolutions can only weaken the current, since in the second series the electromotive power increases as in the first, with the increase of the number of convolutions; while, on the contrary, the resistance is greater in the two series than double the same in one series, on account of the enlarged diameter.

But there is one phænomenon of electro-magnetism to which all the above positions however cannot yet be applied, namely, to the production of the spark. This occurs then only, when the metallic conductor of the current is disturbed at some place; there enters therefore into the circular passage of the current an intermediate conductor, whose length is almost indefinitely small, but whose resistance is almost indefinitely great. We must therefore, in order to apply the above-developed formulæ, first be in a condition to reduce this intermediate conductor to a certain length of wire, with the diameter of the wire given, and thus to determine $m$;—but for this reduction we are yet in want of the data.

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INDEX.