Elements of Algebra

512 ADDITIONS. CHAP. II. $Ap^2 + Bpq + cq^2$, we shall only have to calculate the numbers $p^0, p', p'', p''', \&c.$ and $q^0, q', q'', q''', \&c.$ (Art. 25), and then to try them instead of $p$ and $q$; but this operation may likewise be dispensed with, if we consider, that the quantities $r^0, r', r'', \&c.$ are nothing but the values of the formula in question, when we successively make $p = p^0, p', p'', \&c.$ and $q = q^0, q', q'', \&c.$ We have, therefore, only to consider which is the least term of the series $r^0, r', r'', \&c.$ which we calculate at the same time with the series, $μ, μ', μ'', \&c.$ and that will be the minimum required; we shall then find the corresponding values of $p$ and $q$ by means of the formulæ above quoted. 34. Now I say, that continuing the series, $r^0, r', r'', \&c.$ we must necessarily arrive at two consecutive terms with different signs; and that then the succeeding terms, also, will all have different signs two by two. For, by the preceding Article, we have $r^0 = A(p^0 - aq^0) \times (p^0 - bq^0)$, $r' = A(p' - aq') \times (p' - bq')$, &c. And, from what we demonstrated in Problem 2, it follows, that the quantities $p^0 - aq^0, p' - aq', p'' - aq'', \&c.$ must have alternate signs, and go on diminishing; therefore, 1st, if $b$ is a negative quantity, the quantities $p^0 - bq^0, p' - bq', \&c.$ will all be positive; consequently, the numbers $r^0, r', r'',$ will all have alternate signs; 2dly, if $b$ is a positive quantity, as the quantities $p'' - aq'', p''' - aq''', \&c.$ and much more the quantities $p''/q'' - a, p'''/q''' - a$, form a series, decreasing to infinity, we shall necessarily arrive at one of these last quantities, as $p'''/q''' - a$, which will be $< (a - b)$, abstracting from the sign, and then all the following, $p^{\text{iv}}/q^{\text{iv}} - a, p^{\text{v}}/q^{\text{v}} - a$, will be so likewise; so that all the quantities $a - b + p'''/q''' - a, a - b + p^{\text{iv}}/q^{\text{iv}} - a, \&c.$ will necessarily have the same sign as the quantity $a - b$; consequently, the quantities $p'''/q''' - b, p^{\text{iv}}/q^{\text{iv}} - b, \&c.$ and these $p''' - bq''', p^{\text{iv}} - bq^{\text{iv}}, \&c.$ to infinity, will all have the same sign; therefore, all the numbers $r''', r^{\text{iv}}$, will have alternate signs. Suppose now, in general, that we have arrived at terms, with alternate signs, in the series $r', r'', r''', \&c.$ and that