Elements of Algebra

CHAP. II. ADDITIONS. 513 $p^\lambda$ is the first of those terms, so that all the terms $p^\lambda$, $p^{\lambda+1}$, $p^{\lambda+2}$, &c. to infinity, are alternately positive and negative; I say that none of those terms can be greater than $E$. If, for example, $p'''$, $p^{iv}$, $p^v$, &c. have all alternate signs, it is evident that the products, two by two, $p'''p^{iv}$, $p^{iv}p^v$, &c. will necessarily be negative; but (by the preceding Article), we have $Q^{iv}Q^{iv} - p'''p^{iv} = E$, $Q^vQ^v - p^{iv}p^v = E$, &c. wherefore the positive numbers, $-p'''p^{iv}$, $-p^{iv}p^v$, will all be less than $E$, or at least not greater than $E$; so that, as the numbers $r'$, $r''$, $r'''$, &c. must be integers, the numbers $r'''$, $r^{iv}$, &c. and, in general, the numbers $p^\lambda$, $p^{\lambda+1}$, &c. abstracting from their signs, can never exceed the number $E$. Hence it follows, also, that the terms $q^{iv}$, $q^v$, &c. and, in general, $q^{\lambda+1}$, $q^{\lambda+2}$, &c. can never be greater than $\sqrt{E}$. Whence it is easy to conclude, that the two series $p^\lambda$, $p^{\lambda+1}$, $p^{\lambda+2}$, &c. and $q^{\lambda+1}$, $q^{\lambda+2}$, &c. though carried to infinity, can never be composed but of a certain number of different terms, those terms being, for the first, only the natural numbers as far as $E$, taken positively, or negatively; and for the second, the natural numbers as far as $\sqrt{E}$, with the intermediate fractions $\frac{1}{2}$, $\frac{3}{2}$, $\frac{5}{2}$, &c. likewise taken positively, or negatively; for it is evident, from the formulae of the preceding Article, that the numbers $q'$, $q''$, $q'''$, &c. will always be integer, when $B$ is even; but that they will each contain the fraction $\frac{1}{2}$, when $B$ is odd. Therefore, continuing the two series $p'$, $p''$, $p'''$, &c. and $q'$, $q''$, $q'''$, &c. it will necessarily happen, that two corresponding terms, as $p^\pi$ and $q^\pi$, will return after a certain interval of terms, the number of which may always be supposed even; for, as the same terms, $p^\pi$ and $q^\pi$, must return together an infinite number of times, because the number of different terms in both series is limited, and consequently also the number of their different combinations, it is evident, that if these two terms always returned, after the interval of an odd number of terms, we should only have to consider their returns alternately, and then the intervals would all be composed of an even number of terms. Denoting, therefore, the number of intermediate terms by $2\rho$, we shall have $p^{\pi+2\rho} = p^\pi$, and $q^{\pi+2\rho} = q^\pi$, and then all the terms $p^\pi$, $p^{\pi+1}$, $p^{\pi+2}$, &c. $q^\pi$, $q^{\pi+1}$, $q^{\pi+2}$, and $\mu^\pi$, $\mu^{\pi+1}$, $\mu^{\pi+2}$, &c. will also return at the end of each interval of $2\rho$ terms. For it is evident, from the formulae given in the preceding Article, for the determination of the numbers, $\mu'$, $\mu''$, $\mu'''$, &c. $q'$, $q''$, $q'''$, &c. and $p'$, $p''$, $p'''$, &c. that, since we shall have $p^{\pi+2\rho} = p^\pi$, and $q^{\pi+2\rho} = q^\pi$, we shall also have LL