Induction-Coil Potentials Page 581

Calling this \(m\), and writing \(a\) for \(1/8\pi^2 L_2 C_2 (1 - k^2)\), we have

\[ n_1^2 = a\left(m + 1 - \sqrt{(m - 1)^2 + 4k^2 m}\right), \]

\[ n_2^2 = a\left(m + 1 + \sqrt{(m - 1)^2 + 4k^2 m}\right). \]

Thus

\[ n_1^2 + n_2^2 = 2a(m + 1), \]

\[ n_1^2 n_2^2 = a^2\left\{(m + 1)^2 - (m - 1)^2 - 4k^2 m\right\} = 4ma^2(1 - k^2), \]

\[ (n_2 - n_1)^2 = 2a(m + 1) - 4a\sqrt{m(1 - k^2)}, \]

\[ \frac{n_1^2 n_2^2}{(n_2 - n_1)^2} = \frac{2ma(1 - k^2)}{m + 1 - 2\sqrt{m(1 - k^2)}}, \]

and the upper limit to \(V_2\) is, for given \(L_{21}\), \(L_2 C_2\), and \(i_0\), proportional to the square root of this quantity.

If we wish to consider the effect of varying the primary capacity only, then the upper limit to \(V_2\) is proportional to the square root of

\[ \frac{m}{m + 1 - 2\sqrt{m(1 - k^2)}}, \]

or if

\[ u = \frac{1}{m} = \frac{L_1 C_1}{L_2 C_2}, \]

to the square root of

\[ \frac{1}{1 + u - 2\sqrt{u(1 - k^2)}}. \]

This expression has a maximum value of \(1/k^2\) at \(u = 1 - k^2\), and is equal to unity when \(u = 0\), i.e. when \(C_1 = 0\). Consequently, if the resistances of the coils were negligible, the secondary potential might be expected to be greatest when the primary capacity \(C_1\) has a value of about \(L_2 C_2 (1 - k^2)/L_1\), and this maximum value would be about \(1/k\) times the value the potential would have if the primary condenser were removed altogether.

It is well known that for a given primary current there is a certain value of the primary capacity that gives a maximum secondary sparking distance with an ordinary interruptor^*, but it has been usual to attribute the inferiority of smaller capacities to their failure to prevent sparking at the interruptor, and the view appears to be generally accepted that with a sufficiently rapid interruption of the primary current

^* Cf. Walter, Wied. Ann. lxii. 1897; Mizuno, Phil. Mag. xlv. 1898.