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.