Page:Scientific Memoirs, Vol. 1 (1837).djvu/641

is attained by the current,

$p_{(\mathrm {maximum} )}={\frac {b^{2}f}{2\left(\pi q+(b+\beta ){\sqrt {\left({\frac {\pi m}{\alpha }}\right)}}\right)}}$

(E.)

This expression again shows:

1. That the maximum of the current stands in direct proportion to $f$, i. e., to the power of the magnet, or rather to the strength of the magnetism which is produced in the armature by the placing on of the magnet, and which again vanishes.

2. The maximum is more powerful for a thick wire than for a slender one, for we can bring its expression to the form

${\frac {f}{{\frac {A}{b^{2}}}+{\frac {B}{b}}}},$

which shows that the whole expression increases with the increase of $b$.

3. The maximum decreases with $q$, i. e. it becomes so much the smaller according to the greatness of the cylinder on which the first series of convolutions is wound, it being assumed that the armature does not on that account become greater.

4. It becomes smaller with the increase of $m$, i. e. the greater the free connecting ends of the spirals are, the smaller is the ultimate attainable maximum of the current.

5. Finally, the maximum increases when $\alpha$ increases, i. e. when the space of the armature upon which a series of convolutions can be wound becomes greater.

We shall consider the power of the current of a single convolution wound round the armature to be the same as $m$, then as soon as we put in the general expression (D.) for the current $n=1$, and $\alpha =b+\beta$, we find

$p_{\mathrm {(a\,convolution)} }={\frac {b^{2}f}{\pi (2q+b+\beta )+{}}}m.$

If we divide the expression for the maximum of the current (E.) by this, we may designate the quotients as the maximum of increase, and find that

the maximum of the increase is

${\frac {2q+(b+\beta )+{\frac {m}{\pi }}}{2q+2(b+\beta ){\sqrt {\left({\frac {m}{\alpha \pi }}\right)}}}}$

(F.)

If I propose to find, for instance, with how many series of convolutions I attain the maximum of the current for my magnet and armature, when I take a length of 850 English inches for the wire of the multiplier and the connecting wires together, I have

$\alpha =1\cdot 6,\quad b+\beta =0\cdot 065\quad {\mbox{(wire No. 4)}}\quad q=0\cdot 335,\quad m=850.$

The formula $n={\sqrt {\frac {m}{\alpha \pi }}}$ gives for $n=13\cdot 07$, and the formula (F.) gives the maximum of increase ${}=114\cdot 8$. We shall obtain therefore the