Elements of Algebra

CHAP. II. ADDITIONS. 511 it will become $\mu''P'' + 2\mu''Q'' + P'$, and so on; so that we shall have $P' = \mu^2 P^0 + 2\mu Q^0 + c$ $P'' = \mu'^2 P' + 2\mu' Q' + P^0$ $P''' = \mu''^2 P'' + 2\mu'' Q'' + P'$ $P^{\text{iv}} = \mu'''^2 P''' + 2\mu''' Q''' + P''$, \&c. By means of these formulæ, therefore, we may continue the several series of numbers, $\mu, \mu', \mu''$; $Q^0, Q', Q''$, and $P^0, P', P''$, \&c. to any length, which, as we see, mutually depend on each other, without its being necessary, at the same time, to calculate the numbers $p^0, p', p''$, \&c. and $q^0, q', q''$, \&c. We may also find the values of $P', P'', P'''$, \&c. by more simple formulæ than the preceding, observing that we have $Q'^2 - r' = (\mu'A + \frac{1}{2}B)^2 - A(\mu'^2A + \mu'B + c) = \frac{1}{4}B^2 - AC$, $Q''^2 - P'P'' = (\mu'P' + Q')^2 - P'(\mu'^2P' + 2\mu'Q' + A) = Q'^2 - AP'$, and so on; that is to say, $Q'^2 - P^0P' = \frac{1}{4}E$ $Q''^2 - P'P'' = \frac{1}{4}E$ $Q'''^2 - P''P''' = \frac{1}{4}E$, \&c. Whence we get $P' = \frac{Q'^2 - \frac{1}{4}E}{P^0}$ $P'' = \frac{Q''^2 - \frac{1}{4}E}{P'}$ $P''' = \frac{Q'''^2 - \frac{1}{4}E}{P''}$, \&c. The numbers $\mu, \mu', \mu''$, \&c. having thus been found, we have (Art. 26), the continued fraction, $\alpha = \mu + \frac{1}{\mu' + \frac{1}{\mu'' + \ldots}}$ and, in order to find the minimum of the formula