Page:Scientific Memoirs, Vol. 1 (1837).djvu/640

In the same manner we find Pi = P^ = ^" ~ « ,r (2 nq + n"- (6 + |3) ) + /« (b + ^) ' " ^ '^

If I differentiate this general expression of the power of the current for $n$ series of convolutions in regard to $n$, I obtain

${\frac {d\cdot p_{n}}{d\,n}}=ab^{2}f{\frac {a\pi (2nq+n^{2}(b+\beta ))+m(b+\beta )-a\pi n(2q+2n(b+\beta ))}{\left[a\pi (2nq+n^{2}(b+\beta ))+m(b+\beta )\right]^{2}}}$

If I put this expression ${}=0$, we have after some reductions

$m-a\pi n^{2}=0,$

consequently

$n={\sqrt {\left({\frac {m}{a\pi }}\right)}}.$

I take here the positive sign of the root, because $n$ according to its nature cannot be negative, and $m,\,\alpha ,\,\pi ,$ are all three positive.

If we further develope ${\frac {d^{2}p_{2}}{d\,n^{2}}}$ and put in the expression found this value of $n={\sqrt {\left({\frac {m}{a\pi }}\right)}}$ we obtain a negative magnitude; consequently this value of $n$ represents a maximum of the current.

From the discovered value of $n$ for the maximum of the current, we can infer

1. That the maximum of the action of the magnet on our spirals, for every thickness of the wire, is attained by the same number of series of convolutions; for $n$ is independent of $b+\beta$.

2. That the longer the free ends of the spirals are, or the greater $m$ is, the greater is the number of the series of convolutions required in order to attain the maximum of the action.

3. That the longer the space $\alpha$ is on which the convolutions may be wound round in one series, the less number of series of convolutions are necessary to produce the greatest current.

4.. That the maximum is independent of $q$, i. e. that it is quite indifferent for the number of series of convolutions necessary to the attainment of the maximum, whether they are immediately wound round the cylinder of iron, or round another cylinder which is placed on the other one.

If we put the above found value $n={\sqrt {\left({\frac {m}{a\pi }}\right)}}$ in the general expression of the power which is contained in the equation (D.), we obtain after some reductions, as the expression for the maximum which