taking instead of m" the complete quotient m”+ m + &c. be some quantity greater than unity, which we shall denote by μ; Now x differs from p" only in 1 ; this will Now 1 is less than μ and q′ is greater than q; hence on both accounts the difference between x and 24 is less than the differ 608. To determine limits to the error made in taking any convergent for the continued fraction. By the preceding article the difference between x and 1 2 (2 + 2) 1 ; this is less than and greater than Since q is greater than 9, the error a fortiori is 1 less than and greater than ; these limits are simpler than those first given, though of course not so close. 609. In order that the error made may be less than a given 1 quantity we have therefore only to form the consecutive con 610. Any convergent is nearer to the continued fraction than any other fraction which has a smaller denominator than the convergent has. Let 22 be the convergent, and a fraction, such that s is 8 less than q'. Let x be the continued fraction, and ascending or descending order of magnitude by Art. 603. Now and therefore the difference of ps and qr would be less than 8 that is, an integer less than a proper fraction, which is im q than 2does; in the latter case differs more from x q 8 Reduce the fractions on the right-hand side to a common denominator; we have then in the numerator pp' (uq + q)2 — qq (μp′ +p)3, or that is, μ2 (pp'q2 — qq′p ́3) + pp'q3 — qq'p3, (μ3p'q' — pq) (pq′ – p′q). The factor μ3p'q- pq is necessarily positive; the factor pq-p'q is positive or negative, according as ? is greater or less pp than ; hence is greater or less than that is, is xq p Չ qx greater or less than x3, according as 2 is greater or less than 5. Find three fractions converging to 3.1416. 6. Find a series of fractions converging to the ratio of 5 hours 48 minutes 51 seconds to 24 hours. 7. If P1, P2, Pa be three consecutive convergents, shew 8. Prove that the numerators of any two consecutive convergents have no common measure greater than unity, and similarly for the denominators. 9. If P1, P2, P3, ..... be successive convergents to a con 10. Shew that the difference between the first convergent and the nth convergent is numerically equal to 12. If μ be the nth quotient in a continued fraction greater than unity, shew that R. denote the nth remainder which occurs in the process of P converting the fraction to a continued fraction, shew that Q 16. In converting a fraction in its lowest terms to a continued fraction, shew that any two consecutive remainders have no common measure greater than unity. XLV. REDUCTION OF A QUADRATIC SURD TO A CONTINUED FRACTION. 612. A quadratic surd cannot be reduced to a terminating continued fraction, because the surd would then be equal to a rational fraction, that is, would be commensurable; we shall see, however, that a quadratic surd can be reduced to a continued fraction which does not terminate; we will first give an example, and then the general theory. Take the square root of 6; In the above process the expression which occurs at the beginning of any line is separated into two parts, the first part being the greatest integer which the expression contains, and the second part the remainder; thus the greatest integer in 6 is 2, we therefore write |