ï¿½; the successive terms of this series, after the first, are alternately positive and negative, and consist of fractions with numerators I and denominators continually increasing.
" Now for the interpolation of the rest, I considered that the denominators I, 3, 5, &c., were in arithmetical progression; and that therefore only the numerical coefficients of the numerators were to be investigated.
He extended the "law of continuity" as stated by Johannes Kepler; regarded the denominators of fractions as powers with negative exponents; and deduced from the quadrature of the parabola y=xm, where m is a positive integer, the area of the curves when m is negative or fractional.
Their numerators are denoted by Pi, P2, their denominators by q,, q2, q3, We have the relations p n = an pn-1 +bn pn-2, qn = angn-1 +bngn-2.
- The numerators and denominators of the convergents to the general continued fraction both satisfy the difference equation un =anu„_,+bnun_2.
Having proceeded so far, I considered that the terms (1 -xx) I, (I- (1 -xx) I, (i -xx)l, &c., that is I, I -x 2, I -2x 2 +x 4, I -3x 2 -} 3x 4 -x 6, &c., might be interpolated in the same manner as the areas generated by them, and for this, nothing more was required than to omit the denominators I, 3, 5, 7, &c., in the terms expressing the areas; that is, the coefficients of the terms of the quantity to be interpolated (1-xx) 4 or (1 - xx) 3/2, or generally (I -xx)"n will m m-n e.g.
This denominator must, if the fractions are in their lowest terms (§ 54), be a multiple of each of the denominators; it is usually most convenient that it should be their L.C.M.
(v) The difference of two successive convergents is the reciprocal of the product of their denominators; e.g.
The numerators and denominators of the successive convergents obey the law p n g n _ l - pn-1qn = (- O n, from which it follows at once that every convergent is in its lowest terms. The other principal properties of the convergents are The odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property.
, p, ,, and for their denominators any assigned quantities ql, q2, q 2, The partial fraction b n /a n corresponding to the n th convergent can be found from the relations pn = anpn -I +bnpn -2 1 qn = anq,i l +bngn-2; and the first two partial quotients are given by b l =pi, a1 = ql, 1)102=1,2, a1a2 + b2= q2.
- In order to deal, by way of comparison or addition or subtraction, with fractions which have different denominators, it is necessary to reduce them to a common denominator.
In the sexagesimal system the numerators of the successive fractions (the denominators of which were the successive powers of 60) were followed by', ", "', ", the denominator not being written.